\input texinfo @c -*-texinfo-*- @c %**start of header @setfilename sagereferencemanualdatabases.info @documentencoding UTF-8 @ifinfo @*Generated by Sphinx 1.4.4.@* @end ifinfo @settitle Sage Reference Manual Databases @defindex ge @paragraphindent 0 @exampleindent 4 @finalout @dircategory Python @direntry * SageReferenceManualDatabases: (sagereferencemanualdatabases.info). The SageReferenceManualDatabases reference manual. @end direntry @definfoenclose strong,`,' @definfoenclose emph,`,' @c %**end of header @copying @quotation Sage Reference Manual: Databases 7.4, Oct 20, 2016 Copyright @copyright{} 2005--2016, The Sage Development Team @end quotation @end copying @titlepage @title Sage Reference Manual Databases @insertcopying @end titlepage @contents @c %** start of user preamble @c %** end of user preamble @ifnottex @node Top @top Sage Reference Manual Databases @insertcopying @end ifnottex @c %**start of body @anchor{index doc}@anchor{c} There are numerous specific mathematical databases either included in Sage or available as optional packages. Also, Sage includes two powerful general database packages. Sage includes the ZOPE object oriented database ZODB, which "is a Python object persistence system. It provides transparent object-oriented persistency." Sage also includes the powerful relational database SQLite, along with a Python interface to SQLite. SQlite is a small C library that implements a self-contained, embeddable, zero-configuration SQL database engine. @itemize - @item Transactions are atomic, consistent, isolated, and durable (ACID) even after system crashes and power failures. @item Zero-configuration - no setup or administration needed. @item Implements most of SQL92. (Features not supported) @item A complete database is stored in a single disk file. @item Database files can be freely shared between machines with different byte orders. @item Supports databases up to 2 tebibytes (2^41 bytes) in size. @item Strings and BLOBs up to 2 gibibytes (2^31 bytes) in size. @item Small code footprint: less than 250KiB fully configured or less than 150KiB with optional features omitted. @item Faster than popular client/server database engines for most common operations. @item Simple, easy to use API. @item TCL bindings included. Bindings for many other languages available separately. @item Well-commented source code with over 95% test coverage. @item Self-contained: no external dependencies. @item Sources are in the public domain. Use for any purpose. @end itemize @c nodoctest @menu * Cremona's tables of elliptic curves:: * The Stein-Watkins table of elliptic curves:: * John Jones's tables of number fields:: * The On-Line Encyclopedia of Integer Sequences (OEIS): The On-Line Encyclopedia of Integer Sequences OEIS. * Local copy of Sloane On-Line Encyclopedia of Integer Sequences:: * FindStat - the Combinatorial Statistic Finder.: FindStat - the Combinatorial Statistic Finder. * Frank Luebeck's tables of Conway polynomials over finite fields:: * Tables of zeros of the Riemann-Zeta function:: * Ideals from the Symbolic Data project:: * Cunningham table:: * Database of Hilbert Polynomials:: * Database of Modular Polynomials:: * Indices and Tables:: * Python Module Index:: * Index:: @end menu @node Cremona's tables of elliptic curves,The Stein-Watkins table of elliptic curves,Top,Top @anchor{sage/databases/cremona cremona-s-tables-of-elliptic-curves}@anchor{d}@anchor{sage/databases/cremona sage-databases-cremona}@anchor{e}@anchor{sage/databases/cremona doc}@anchor{f}@anchor{sage/databases/cremona databases}@anchor{10} @chapter Cremona's tables of elliptic curves @c This file has been autogenerated. @anchor{sage/databases/cremona module-sage databases cremona}@anchor{1} @geindex sage.databases.cremona (module) Sage includes John Cremona's tables of elliptic curves in an easy-to-use format. An instance of the class CremonaDatabase() gives access to the database. If the optional full CremonaDatabase is not installed, a mini-version is included by default with Sage. It contains Weierstrass equations, rank, and torsion for curves up to conductor 10000. The large database includes all curves in John Cremona's tables. It also includes data related to the BSD conjecture and modular degrees for all of these curves, and generators for the Mordell-Weil groups. To install it, run the following in the shell: @example sage -i database_cremona_ellcurve @end example This causes the latest version of the database to be downloaded from the internet. Both the mini and full versions of John Cremona's tables are stored in SAGE_SHARE/cremona as SQLite databases. The mini version has the layout: @example CREATE TABLE t_class(conductor INTEGER, class TEXT PRIMARY KEY, rank INTEGER); CREATE TABLE t_curve(class TEXT, curve TEXT PRIMARY KEY, eqn TEXT UNIQUE, tors INTEGER); CREATE INDEX i_t_class_conductor ON t_class(conductor); CREATE INDEX i_t_curve_class ON t_curve(class); @end example while the full version has the layout: @example CREATE TABLE t_class(conductor INTEGER, class TEXT PRIMARY KEY, rank INTEGER, L REAL, deg INTEGER); CREATE TABLE t_curve(class TEXT, curve TEXT PRIMARY KEY, eqn TEXT UNIQUE, gens TEXT, tors INTEGER, cp INTEGER, om REAL, reg REAL, sha); CREATE INDEX i_t_class_conductor ON t_class(conductor); CREATE INDEX i_t_curve_class ON t_curve(class); @end example @geindex CremonaDatabase() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona CremonaDatabase}@anchor{11} @deffn {Function} sage.databases.cremona.CremonaDatabase (name=None, mini=None, set_global=None) Initializes the Cremona database with name @code{name}. If @code{name} is @code{None} it instead initializes large Cremona database (named 'cremona'), if available or default mini Cremona database (named 'cremona mini'). If the Cremona database in question is in the format of the mini database, you must set @code{mini=True}, otherwise it must be set to @code{False}. If you would like other components of Sage to use this database, mark @code{set_global=True}. TESTS: @example sage: c = CremonaDatabase() sage: isinstance(c, sage.databases.cremona.MiniCremonaDatabase) True sage: isinstance(c, sage.databases.cremona.LargeCremonaDatabase) # optional - database_cremona_ellcurve True @end example Verify that trac ticket #12341@footnote{https://trac.sagemath.org/12341} has been resolved: @example sage: c = CremonaDatabase('should not exist',mini=True) Traceback (most recent call last): ... ValueError: Desired database (='should not exist') does not exist sage: c = CremonaDatabase('should not exist',mini=False) Traceback (most recent call last): ... ValueError: Desired database (='should not exist') does not exist sage: from sage.env import SAGE_SHARE sage: os.path.isfile(os.path.join(SAGE_SHARE,'cremona','should_not_exist.db')) False @end example @end deffn @geindex LargeCremonaDatabase (class in sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona LargeCremonaDatabase}@anchor{12} @deffn {Class} sage.databases.cremona.LargeCremonaDatabase (name, read_only=True, build=False) Bases: @ref{13,,sage.databases.cremona.MiniCremonaDatabase} The Cremona database of elliptic curves. EXAMPLES: @example sage: c = CremonaDatabase('cremona') # optional - database_cremona_ellcurve sage: c.allcurves(11) # optional - database_cremona_ellcurve @{'a1': [[0, -1, 1, -10, -20], 0, 5], 'a2': [[0, -1, 1, -7820, -263580], 0, 1], 'a3': [[0, -1, 1, 0, 0], 0, 5]@} @end example @geindex allbsd() (sage.databases.cremona.LargeCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona LargeCremonaDatabase allbsd}@anchor{14} @deffn {Method} allbsd (N) Return the allbsd table for conductor N. The entries are: @example [id, tamagawa_product, Omega_E, L, Reg_E, Sha_an(E)] @end example where id is the isogeny class (letter) followed by a number, e.g., b3, and L is @math{L^r(E@comma{}1)/r!}, where E has rank r. INPUT: @itemize - @item @code{N} - int, the conductor @end itemize OUTPUT: dict containing the allbsd table for each isogeny class in conductor N EXAMPLES: @example sage: c = CremonaDatabase() sage: c.allbsd(12) # optional - database_cremona_ellcurve @{@} sage: c.allbsd(19)['a3'] # optional - database_cremona_ellcurve [1, 4.07927920046493, 0.453253244496104, 1.0, 1] sage: c.allbsd(12001)['a1'] # optional - database_cremona_ellcurve [2, 3.27608135248722, 1.54910143090506, 0.236425971187952, 1.0] @end example @end deffn @geindex allgens() (sage.databases.cremona.LargeCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona LargeCremonaDatabase allgens}@anchor{15} @deffn {Method} allgens (N) Return the allgens table for conductor N. INPUT: @itemize - @item @code{N} - int, the conductor @end itemize OUTPUT: @itemize - @item @code{dict} - id:[points, ...], ... @end itemize EXAMPLES: @example sage: c = CremonaDatabase() sage: c.allgens(12) # optional - database_cremona_ellcurve @{@} sage: c.allgens(1001)['a1'] # optional - database_cremona_ellcurve [[61, 181, 1]] sage: c.allgens(12001)['a1'] # optional - database_cremona_ellcurve [[7, 2, 1]] @end example @end deffn @geindex degphi() (sage.databases.cremona.LargeCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona LargeCremonaDatabase degphi}@anchor{16} @deffn {Method} degphi (N) Return the degphi table for conductor N. INPUT: @itemize - @item @code{N} - int, the conductor @end itemize OUTPUT: @itemize - @item @code{dict} - id:degphi, ... @end itemize EXAMPLES: @example sage: c = CremonaDatabase() sage: c.degphi(11) # optional - database_cremona_ellcurve @{'a1': 1@} sage: c.degphi(12001)['c1'] # optional - database_cremona_ellcurve 1640 @end example @end deffn @end deffn @geindex MiniCremonaDatabase (class in sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase}@anchor{13} @deffn {Class} sage.databases.cremona.MiniCremonaDatabase (name, read_only=True, build=False) Bases: sage.databases.sql_db.SQLDatabase@footnote{../../../../../../html/en/reference/misc/sage/databases/sql_db.html#sage.databases.sql_db.SQLDatabase} The Cremona database of elliptic curves. EXAMPLES: @example sage: c = CremonaDatabase() sage: c.allcurves(11) @{'a1': [[0, -1, 1, -10, -20], 0, 5], 'a2': [[0, -1, 1, -7820, -263580], 0, 1], 'a3': [[0, -1, 1, 0, 0], 0, 5]@} @end example @geindex allcurves() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase allcurves}@anchor{17} @deffn {Method} allcurves (N) Returns the allcurves table of curves of conductor N. INPUT: @itemize - @item @code{N} - int, the conductor @end itemize OUTPUT: @itemize - @item @code{dict} - id:[ainvs, rank, tor], ... @end itemize EXAMPLES: @example sage: c = CremonaDatabase() sage: c.allcurves(11)['a3'] [[0, -1, 1, 0, 0], 0, 5] sage: c.allcurves(12) @{@} sage: c.allcurves(12001)['a1'] # optional - database_cremona_ellcurve [[1, 0, 0, -101, 382], 1, 1] @end example @end deffn @geindex coefficients_and_data() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase coefficients_and_data}@anchor{18} @deffn {Method} coefficients_and_data (label) Return the Weierstrass coefficients and other data for the curve with given label. EXAMPLES: @example sage: c, d = CremonaDatabase().coefficients_and_data('144b1') sage: c [0, 0, 0, 6, 7] sage: d['conductor'] 144 sage: d['cremona_label'] '144b1' sage: d['rank'] 0 sage: d['torsion_order'] 2 @end example Check that trac ticket #17904@footnote{https://trac.sagemath.org/17904} is fixed: @example sage: 'gens' in CremonaDatabase().coefficients_and_data('100467a2')[1] # optional - database_cremona_ellcurve True @end example @end deffn @geindex conductor_range() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase conductor_range}@anchor{19} @deffn {Method} conductor_range () Return the range of conductors that are covered by the database. OUTPUT: tuple of ints (N1,N2+1) where N1 is the smallest and N2 the largest conductor for which the database is complete. EXAMPLES: @example sage: c = CremonaDatabase('cremona mini') sage: c.conductor_range() (1, 10000) @end example @end deffn @geindex curves() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase curves}@anchor{1a} @deffn {Method} curves (N) Returns the curves table of all @emph{optimal} curves of conductor N. INPUT: @itemize - @item @code{N} - int, the conductor @end itemize OUTPUT: @itemize - @item @code{dict} - id:[ainvs, rank, tor], ... @end itemize EXAMPLES: Optimal curves of conductor 37: @example sage: CremonaDatabase().curves(37) @{'a1': [[0, 0, 1, -1, 0], 1, 1], 'b1': [[0, 1, 1, -23, -50], 0, 3]@} @end example Note the 'h3', which is the unique case in the tables where the optimal curve doesn't have label ending in 1: @example sage: list(sorted(CremonaDatabase().curves(990).keys())) ['a1', 'b1', 'c1', 'd1', 'e1', 'f1', 'g1', 'h3', 'i1', 'j1', 'k1', 'l1'] @end example TESTS: @example sage: c = CremonaDatabase() sage: c.curves(12001)['a1'] # optional - database_cremona_ellcurve [[1, 0, 0, -101, 382], 1, 1] @end example @end deffn @geindex data_from_coefficients() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase data_from_coefficients}@anchor{1b} @deffn {Method} data_from_coefficients (ainvs) Return elliptic curve data for the curve with given Weierstrass coefficients. EXAMPLES: @example sage: d = CremonaDatabase().data_from_coefficients([1, -1, 1, 31, 128]) sage: d['conductor'] 1953 sage: d['cremona_label'] '1953c1' sage: d['rank'] 1 sage: d['torsion_order'] 2 @end example Check that trac ticket #17904@footnote{https://trac.sagemath.org/17904} is fixed: @example sage: ai = EllipticCurve('100467a2').ainvs() # optional - database_cremona_ellcurve sage: 'gens' in CremonaDatabase().data_from_coefficients(ai) # optional - database_cremona_ellcurve True @end example @end deffn @geindex elliptic_curve() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase elliptic_curve}@anchor{1c} @deffn {Method} elliptic_curve (label) Return an elliptic curve with given label with some data about it from the database pre-filled in. INPUT: @itemize - @item @code{label} - str (Cremona or LMFDB label) @end itemize OUTPUT: @itemize - @item an sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field@footnote{../../../../../../html/en/reference/curves/sage/schemes/elliptic_curves/ell_rational_field.html#sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field} @end itemize @cartouche @quotation Note For more details on LMFDB labels see @ref{1d,,parse_lmfdb_label()}. @end quotation @end cartouche EXAMPLES: @example sage: c = CremonaDatabase() sage: c.elliptic_curve('11a1') Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field sage: c.elliptic_curve('12001a1') # optional - database_cremona_ellcurve Elliptic Curve defined by y^2 + x*y = x^3 - 101*x + 382 over Rational Field sage: c.elliptic_curve('48c1') Traceback (most recent call last): ... ValueError: There is no elliptic curve with label 48c1 in the database @end example You can also use LMFDB labels: @example sage: c.elliptic_curve('462.f3') Elliptic Curve defined by y^2 + x*y = x^3 - 363*x + 1305 over Rational Field @end example @end deffn @geindex elliptic_curve_from_ainvs() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase elliptic_curve_from_ainvs}@anchor{1e} @deffn {Method} elliptic_curve_from_ainvs (ainvs) Returns the elliptic curve in the database of with minimal ainvs, if it exists, or raises a RuntimeError exception otherwise. INPUT: @itemize - @item @code{ainvs} - list (5-tuple of int's); the minimal Weierstrass model for an elliptic curve @end itemize OUTPUT: EllipticCurve EXAMPLES: @example sage: c = CremonaDatabase() sage: c.elliptic_curve_from_ainvs([0, -1, 1, -10, -20]) Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field sage: c.elliptic_curve_from_ainvs([1, 0, 0, -101, 382]) # optional - database_cremona_ellcurve Elliptic Curve defined by y^2 + x*y = x^3 - 101*x + 382 over Rational Field @end example Old (pre-2006) Cremona labels are also allowed: @example sage: c.elliptic_curve('9450KKKK1') Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 5*x + 7 over Rational Field @end example Make sure trac ticket #12565@footnote{https://trac.sagemath.org/12565} is fixed: @example sage: c.elliptic_curve('10a1') Traceback (most recent call last): ... ValueError: There is no elliptic curve with label 10a1 in the database @end example @end deffn @geindex isogeny_class() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase isogeny_class}@anchor{1f} @deffn {Method} isogeny_class (label) Returns the isogeny class of elliptic curves that are isogenous to the curve with given Cremona label. INPUT: @itemize - @item @code{label} - string @end itemize OUTPUT: @itemize - @item @code{list} - list of EllipticCurve objects. @end itemize EXAMPLES: @example sage: c = CremonaDatabase() sage: c.isogeny_class('11a1') [Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 - x^2 over Rational Field] sage: c.isogeny_class('12001a1') # optional - database_cremona_ellcurve [Elliptic Curve defined by y^2 + x*y = x^3 - 101*x + 382 over Rational Field] @end example @end deffn @geindex isogeny_classes() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase isogeny_classes}@anchor{20} @deffn {Method} isogeny_classes (conductor) Return the allcurves data (ainvariants, rank and torsion) for the elliptic curves in the database of given conductor as a list of lists, one for each isogeny class. The curve with number 1 is always listed first. EXAMPLES: @example sage: c = CremonaDatabase() sage: c.isogeny_classes(11) [[[[0, -1, 1, -10, -20], 0, 5], [[0, -1, 1, -7820, -263580], 0, 1], [[0, -1, 1, 0, 0], 0, 5]]] sage: c.isogeny_classes(12001) # optional - database_cremona_ellcurve [[[[1, 0, 0, -101, 382], 1, 1]], [[[0, 0, 1, -247, 1494], 1, 1]], [[[0, 0, 1, -4, -18], 1, 1]], [[[0, 1, 1, -10, 18], 1, 1]]] @end example @end deffn @geindex iter() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase iter}@anchor{21} @deffn {Method} iter (conductors) Return an iterator through all curves in the database with given conductors. INPUT: @itemize - @item @code{conductors} - list or generator of ints @end itemize OUTPUT: generator that iterates over EllipticCurve objects. EXAMPLES: @example sage: [e.cremona_label() for e in CremonaDatabase().iter([11..15])] ['11a1', '11a2', '11a3', '14a1', '14a2', '14a3', '14a4', '14a5', '14a6', '15a1', '15a2', '15a3', '15a4', '15a5', '15a6', '15a7', '15a8'] @end example @end deffn @geindex iter_optimal() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase iter_optimal}@anchor{22} @deffn {Method} iter_optimal (conductors) Return an iterator through all optimal curves in the database with given conductors. INPUT: @itemize - @item @code{conductors} - list or generator of ints @end itemize OUTPUT: generator that iterates over EllipticCurve objects. EXAMPLES: We list optimal curves with conductor up to 20: @example sage: [e.cremona_label() for e in CremonaDatabase().iter_optimal([11..20])] ['11a1', '14a1', '15a1', '17a1', '19a1', '20a1'] @end example Note the unfortunate 990h3 special case: @example sage: [e.cremona_label() for e in CremonaDatabase().iter_optimal([990])] ['990a1', '990b1', '990c1', '990d1', '990e1', '990f1', '990g1', '990h3', '990i1', '990j1', '990k1', '990l1'] @end example @end deffn @geindex largest_conductor() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase largest_conductor}@anchor{23} @deffn {Method} largest_conductor () The largest conductor for which the database is complete. OUTPUT: @itemize - @item @code{int} - largest conductor @end itemize EXAMPLES: @example sage: c = CremonaDatabase('cremona mini') sage: c.largest_conductor() 9999 @end example @end deffn @geindex list() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase list}@anchor{24} @deffn {Method} list (conductors) Returns a list of all curves with given conductors. INPUT: @itemize - @item @code{conductors} - list or generator of ints @end itemize OUTPUT: @itemize - @item list of EllipticCurve objects. @end itemize EXAMPLES: @example sage: CremonaDatabase().list([37]) [Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 3*x + 1 over Rational Field] @end example @end deffn @geindex list_optimal() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase list_optimal}@anchor{25} @deffn {Method} list_optimal (conductors) Returns a list of all optimal curves with given conductors. INPUT: @itemize - @item @table @asis @item @code{conductors} - list or generator of ints list of EllipticCurve objects. @end table @end itemize OUTPUT: list of EllipticCurve objects. EXAMPLES: @example sage: CremonaDatabase().list_optimal([37]) [Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field] @end example @end deffn @geindex number_of_curves() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase number_of_curves}@anchor{26} @deffn {Method} number_of_curves (N=0, i=0) Returns the number of curves stored in the database with conductor N. If N = 0, returns the total number of curves in the database. If i is nonzero, returns the number of curves in the i-th isogeny class. If i is a Cremona letter code, e.g., 'a' or 'bc', it is converted to the corresponding number. INPUT: @itemize - @item @code{N} - int @item @code{i} - int or str @end itemize OUTPUT: int EXAMPLES: @example sage: c = CremonaDatabase() sage: c.number_of_curves(11) 3 sage: c.number_of_curves(37) 4 sage: c.number_of_curves(990) 42 sage: num = c.number_of_curves() @end example @end deffn @geindex number_of_isogeny_classes() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase number_of_isogeny_classes}@anchor{27} @deffn {Method} number_of_isogeny_classes (N=0) Returns the number of isogeny classes of curves in the database of conductor N. If N is 0, return the total number of isogeny classes of curves in the database. INPUT: @itemize - @item @code{N} - int @end itemize OUTPUT: int EXAMPLES: @example sage: c = CremonaDatabase() sage: c.number_of_isogeny_classes(11) 1 sage: c.number_of_isogeny_classes(37) 2 sage: num = c.number_of_isogeny_classes() @end example @end deffn @geindex random() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase random}@anchor{28} @deffn {Method} random () Returns a random curve from the database. EXAMPLES: @example sage: CremonaDatabase().random() # random -- depends on database installed Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 224*x + 3072 over Rational Field @end example @end deffn @geindex smallest_conductor() (sage.databases.cremona.MiniCremonaDatabase method) @anchor{sage/databases/cremona sage databases cremona MiniCremonaDatabase smallest_conductor}@anchor{29} @deffn {Method} smallest_conductor () The smallest conductor for which the database is complete: always 1. OUTPUT: @itemize - @item @code{int} - smallest conductor @end itemize @cartouche @quotation Note This always returns the integer 1, since that is the smallest conductor for which the database is complete, although there are no elliptic curves of conductor 1. The smallest conductor of a curve in the database is 11. @end quotation @end cartouche EXAMPLES: @example sage: CremonaDatabase().smallest_conductor() 1 @end example @end deffn @end deffn @geindex build() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona build}@anchor{2a} @deffn {Function} sage.databases.cremona.build (name, data_tgz, largest_conductor=0, mini=False, decompress=True) Build the CremonaDatabase with given name from scratch using the data_tgz tarball. @cartouche @quotation Note For data up to level 350000, this function takes about 3m40s. The resulting database occupies 426MB disk space. @end quotation @end cartouche To create the large Cremona database from Cremona's data_tgz tarball, obtainable from @indicateurl{http://homepages.warwick.ac.uk/staff/J.E.Cremona/ftp/data/}, run the following command: @example sage: d = sage.databases.cremona.build('cremona','ecdata.tgz') # not tested @end example @end deffn @geindex class_to_int() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona class_to_int}@anchor{2b} @deffn {Function} sage.databases.cremona.class_to_int (k) Converts class id string into an integer. Note that this is the inverse of cremona_letter_code. EXAMPLES: @example sage: import sage.databases.cremona as cremona sage: cremona.class_to_int('ba') 26 sage: cremona.class_to_int('cremona') 821863562 sage: cremona.cremona_letter_code(821863562) 'cremona' @end example @end deffn @geindex cremona_letter_code() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona cremona_letter_code}@anchor{2c} @deffn {Function} sage.databases.cremona.cremona_letter_code (n) Returns the Cremona letter code corresponding to an integer. For example, 0 - a 25 - z 26 - ba 51 - bz 52 - ca 53 - cb etc. @cartouche @quotation Note This is just the base 26 representation of n, where a=0, b=1, ..., z=25. This extends the old Cremona notation (counting from 0) for the first 26 classes, and is different for classes above 26. @end quotation @end cartouche INPUT: @itemize - @item @code{n} (int) -- a non-negative integer @end itemize OUTPUT: str EXAMPLES: @example sage: from sage.databases.cremona import cremona_letter_code sage: cremona_letter_code(0) 'a' sage: cremona_letter_code(26) 'ba' sage: cremona_letter_code(27) 'bb' sage: cremona_letter_code(521) 'ub' sage: cremona_letter_code(53) 'cb' sage: cremona_letter_code(2005) 'czd' @end example TESTS: @example sage: cremona_letter_code(QQ) Traceback (most recent call last): ... ValueError: Cremona letter codes are only defined for non-negative integers sage: cremona_letter_code(x) Traceback (most recent call last): ... ValueError: Cremona letter codes are only defined for non-negative integers sage: cremona_letter_code(-1) Traceback (most recent call last): ... ValueError: Cremona letter codes are only defined for non-negative integers sage: cremona_letter_code(3.14159) Traceback (most recent call last): ... ValueError: Cremona letter codes are only defined for non-negative integers @end example @end deffn @geindex cremona_to_lmfdb() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona cremona_to_lmfdb}@anchor{2d} @deffn {Function} sage.databases.cremona.cremona_to_lmfdb (cremona_label, CDB=None) Converts a Cremona label into an LMFDB label. See @ref{1d,,parse_lmfdb_label()} for an explanation of LMFDB labels. INPUT: @itemize - @item @code{cremona_label} -- a string, the Cremona label of a curve. This can be the label of a curve (e.g. '990j1') or of an isogeny class (e.g. '990j') @item @code{CDB} -- the Cremona database in which to look up the isogeny classes of the same conductor. @end itemize OUTPUT: @itemize - @item @code{lmfdb_label} -- a string, the corresponding LMFDB label. @end itemize EXAMPLES: @example sage: from sage.databases.cremona import cremona_to_lmfdb, lmfdb_to_cremona sage: cremona_to_lmfdb('990j1') '990.h3' sage: lmfdb_to_cremona('990.h3') '990j1' @end example TESTS: @example sage: for label in ['5077a1','66a3','102b','420c2']: ... assert(lmfdb_to_cremona(cremona_to_lmfdb(label)) == label) sage: for label in ['438.c2','306.b','462.f3']: ... assert(cremona_to_lmfdb(lmfdb_to_cremona(label)) == label) @end example @end deffn @geindex is_optimal_id() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona is_optimal_id}@anchor{2e} @deffn {Function} sage.databases.cremona.is_optimal_id (id) Returns true if the Cremona id refers to an optimal curve, and false otherwise. The curve is optimal if the id, which is of the form [letter code][number] has number 1. @cartouche @quotation Note 990h3 is the optimal curve in that class, so doesn't obey this rule. @end quotation @end cartouche INPUT: @itemize - @item @code{id} - str of form letter code followed by an integer, e.g., a3, bb5, etc. @end itemize OUTPUT: bool EXAMPLES: @example sage: from sage.databases.cremona import is_optimal_id sage: is_optimal_id('b1') True sage: is_optimal_id('bb1') True sage: is_optimal_id('c1') True sage: is_optimal_id('c2') False @end example @end deffn @geindex lmfdb_to_cremona() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona lmfdb_to_cremona}@anchor{2f} @deffn {Function} sage.databases.cremona.lmfdb_to_cremona (lmfdb_label, CDB=None) Converts an LMFDB labe into a Cremona label. See @ref{1d,,parse_lmfdb_label()} for an explanation of LMFDB labels. INPUT: @itemize - @item @code{lmfdb_label} -- a string, the LMFDB label of a curve. This can be the label of a curve (e.g. '990.j1') or of an isogeny class (e.g. '990.j') @item @code{CDB} -- the Cremona database in which to look up the isogeny classes of the same conductor. @end itemize OUTPUT: @itemize - @item @code{cremona_label} -- a string, the corresponding Cremona label. @end itemize EXAMPLES: @example sage: from sage.databases.cremona import cremona_to_lmfdb, lmfdb_to_cremona sage: lmfdb_to_cremona('990.h3') '990j1' sage: cremona_to_lmfdb('990j1') '990.h3' @end example @end deffn @geindex old_cremona_letter_code() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona old_cremona_letter_code}@anchor{30} @deffn {Function} sage.databases.cremona.old_cremona_letter_code (n) Returns the @emph{old} Cremona letter code corresponding to an integer. integer. For example: @example 1 --> A 26 --> Z 27 --> AA 52 --> ZZ 53 --> AAA etc. @end example INPUT: @itemize - @item @code{n} - int @end itemize OUTPUT: str EXAMPLES: @example sage: from sage.databases.cremona import old_cremona_letter_code sage: old_cremona_letter_code(1) 'A' sage: old_cremona_letter_code(26) 'Z' sage: old_cremona_letter_code(27) 'AA' sage: old_cremona_letter_code(521) 'AAAAAAAAAAAAAAAAAAAAA' sage: old_cremona_letter_code(53) 'AAA' sage: old_cremona_letter_code(2005) 'CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC' @end example @end deffn @geindex parse_cremona_label() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona parse_cremona_label}@anchor{31} @deffn {Function} sage.databases.cremona.parse_cremona_label (label) Given a Cremona label that defines an elliptic curve, e.g., 11a1 or 37b3, parse the label and return the conductor, isogeny class label, and number. For this function, the curve number may be omitted, in which case it defaults to 1. If the curve number and isogeny class are both omitted (label is just a string representing a conductor), then the isogeny class defaults to 'a' and the number to 1. Valid labels consist of one or more digits, followed by zero or more letters (either all in upper case for an old Cremona label, or all in lower case), followed by zero or more digits. INPUT: @itemize - @item @code{label} - str @end itemize OUTPUT: @itemize - @item @code{int} - the conductor @item @code{str} - the isogeny class label @item @code{int} - the number @end itemize EXAMPLES: @example sage: from sage.databases.cremona import parse_cremona_label sage: parse_cremona_label('37a2') (37, 'a', 2) sage: parse_cremona_label('37b1') (37, 'b', 1) sage: parse_cremona_label('10bb2') (10, 'bb', 2) sage: parse_cremona_label('11a') (11, 'a', 1) sage: parse_cremona_label('11') (11, 'a', 1) @end example Valid old Cremona labels are allowed: @example sage: parse_cremona_label('17CCCC') (17, 'dc', 1) sage: parse_cremona_label('5AB2') Traceback (most recent call last): ... ValueError: 5AB2 is not a valid Cremona label @end example TESTS: @example sage: from sage.databases.cremona import parse_cremona_label sage: parse_cremona_label('x11') Traceback (most recent call last): ... ValueError: x11 is not a valid Cremona label @end example @end deffn @geindex parse_lmfdb_label() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona parse_lmfdb_label}@anchor{1d} @deffn {Function} sage.databases.cremona.parse_lmfdb_label (label) Given an LMFDB label that defines an elliptic curve, e.g., 11.a1 or 37.b3, parse the label and return the conductor, isogeny class label, and number. The LMFDB label (named after the L-functions and modular forms database), is determined by the following two orders: @itemize - @item Isogeny classes with the same conductor are ordered lexicographically by the coefficients in the q-expansion of the associated modular form. @item Curves within the same isogeny class are ordered lexicographically by the a-invariants of the minimal model. @end itemize The format is ., where the isogeny class is encoded using the same base-26 encoding into letters used in Cremona's labels. For example, 990.h3 is the same as Cremona's 990j1 For this function, the curve number may be omitted, in which case it defaults to 1. If the curve number and isogeny class are both omitted (label is just a string representing a conductor), then the isogeny class defaults to 'a' and the number to 1. INPUT: @itemize - @item @code{label} - str @end itemize OUTPUT: @itemize - @item @code{int} - the conductor @item @code{str} - the isogeny class label @item @code{int} - the number @end itemize EXAMPLES: @example sage: from sage.databases.cremona import parse_lmfdb_label sage: parse_lmfdb_label('37.a2') (37, 'a', 2) sage: parse_lmfdb_label('37.b') (37, 'b', 1) sage: parse_lmfdb_label('10.bb2') (10, 'bb', 2) @end example @end deffn @geindex sort_key() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona sort_key}@anchor{32} @deffn {Function} sage.databases.cremona.sort_key (key1) Comparison key for curve id strings. @cartouche @quotation Note Not the same as standard lexicographic order! @end quotation @end cartouche EXAMPLES: @example sage: from sage.databases.cremona import sort_key sage: l = ['ba1', 'z1'] sage: sorted(l, key=sort_key) ['z1', 'ba1'] @end example @end deffn @geindex split_code() (in module sage.databases.cremona) @anchor{sage/databases/cremona sage databases cremona split_code}@anchor{33} @deffn {Function} sage.databases.cremona.split_code (key) Splits class+curve id string into its two parts. EXAMPLES: @example sage: import sage.databases.cremona as cremona sage: cremona.split_code('ba2') ('ba', '2') @end example @end deffn @c nodoctest @node The Stein-Watkins table of elliptic curves,John Jones's tables of number fields,Cremona's tables of elliptic curves,Top @anchor{sage/databases/stein_watkins the-stein-watkins-table-of-elliptic-curves}@anchor{34}@anchor{sage/databases/stein_watkins sage-databases-stein-watkins}@anchor{35}@anchor{sage/databases/stein_watkins doc}@anchor{36} @chapter The Stein-Watkins table of elliptic curves @c This file has been autogenerated. @anchor{sage/databases/stein_watkins module-sage databases stein_watkins}@anchor{a} @geindex sage.databases.stein_watkins (module) Sage gives access to the Stein-Watkins table of elliptic curves, via an optional package that you must install. This is a huge database of elliptic curves. You can install the database (a 2.6GB package) with the command @example sage -i database_stein_watkins @end example You can also automatically download a small version, which takes much less time, using the command @example sage -i database_stein_watkins_mini @end example This database covers a wide range of conductors, but unlike the @ref{1,,Cremona database}, this database need not list all curves of a given conductor. It lists the curves whose coefficients are not "too large" (see @ref{37,,[SteinWatkins]}). @itemize - @item The command @code{SteinWatkinsAllData(n)} returns an iterator over the curves in the @math{n}-th Stein-Watkins table, which contains elliptic curves of conductor between @math{n10^5} and @math{(n+1)10^5}. Here @math{n} can be between 0 and 999, inclusive. @item The command @code{SteinWatkinsPrimeData(n)} returns an iterator over the curves in the @math{n^@{th@}} Stein-Watkins prime table, which contains prime conductor elliptic curves of conductor between @math{n10^6} and @math{(n+1)10^6}. Here @math{n} varies between 0 and 99, inclusive. @end itemize EXAMPLES: We obtain the first table of elliptic curves. @example sage: d = SteinWatkinsAllData(0) sage: d Stein-Watkins Database a.0 Iterator @end example We type @code{next(d)} to get each isogeny class of curves from @code{d}: @example sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 11 sage: next(d) # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 14 sage: next(d) # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 15 @end example An isogeny class has a number of attributes that give data about the isogeny class, such as the rank, equations of curves, conductor, leading coefficient of @math{L}-function, etc. @example sage: C.data # optional - database_stein_watkins ['11', '[11]', '0', '0.253842', '25', '+*1'] sage: C.curves # optional - database_stein_watkins [[[0, -1, 1, 0, 0], '(1)', '1', '5'], [[0, -1, 1, -10, -20], '(5)', '1', '5'], [[0, -1, 1, -7820, -263580], '(1)', '1', '1']] sage: C.conductor # optional - database_stein_watkins 11 sage: C.leading_coefficient # optional - database_stein_watkins '0.253842' sage: C.modular_degree # optional - database_stein_watkins '+*1' sage: C.rank # optional - database_stein_watkins 0 sage: C.isogeny_number # optional - database_stein_watkins '25' @end example If we were to continue typing @code{next(d)} we would iterate over all curves in the Stein-Watkins database up to conductor @math{10^5}. We could also type @code{for C in d: ...} To access the data file starting at @math{10^5} do the following: @example sage: d = SteinWatkinsAllData(1) sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 100002 sage: C.curves # optional - database_stein_watkins [[[1, 1, 0, 112, 0], '(8,1,2,1)', 'X', '2'], [[1, 1, 0, -448, -560], '[4,2,1,2]', 'X', '2']] @end example Next we access the prime-conductor data: @example sage: d = SteinWatkinsPrimeData(0) sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 11 @end example Each call @code{next(d)} gives another elliptic curve of prime conductor: @example sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 17 sage: C.curves # optional - database_stein_watkins [[[1, -1, 1, -1, 0], '[1]', '1', '4'], [[1, -1, 1, -6, -4], '[2]', '1', '2x'], [[1, -1, 1, -1, -14], '(4)', '1', '4'], [[1, -1, 1, -91, -310], '[1]', '1', '2']] sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 19 @end example REFERENCE: @anchor{sage/databases/stein_watkins steinwatkins}@anchor{37}@w{(SteinWatkins)} William Stein and Mark Watkins, @emph{A database of elliptic curves---first report}. In @emph{Algorithmic number theory (ANTS V), Sydney, 2002}, Lecture Notes in Computer Science 2369, Springer, 2002, p267--275. @indicateurl{http://modular.math.washington.edu/papers/stein-watkins/} @geindex SteinWatkinsAllData (class in sage.databases.stein_watkins) @anchor{sage/databases/stein_watkins sage databases stein_watkins SteinWatkinsAllData}@anchor{38} @deffn {Class} sage.databases.stein_watkins.SteinWatkinsAllData (num) Class for iterating through one of the Stein-Watkins database files for all conductors. @geindex iter_levels() (sage.databases.stein_watkins.SteinWatkinsAllData method) @anchor{sage/databases/stein_watkins sage databases stein_watkins SteinWatkinsAllData iter_levels}@anchor{39} @deffn {Method} iter_levels () Iterate through the curve classes, but grouped into lists by level. EXAMPLE: @example sage: d = SteinWatkinsAllData(1) sage: E = d.iter_levels() sage: next(E) # optional - database_stein_watkins [Stein-Watkins isogeny class of conductor 100002] sage: next(E) # optional - database_stein_watkins [Stein-Watkins isogeny class of conductor 100005, Stein-Watkins isogeny class of conductor 100005] sage: next(E) # optional - database_stein_watkins [Stein-Watkins isogeny class of conductor 100007] @end example @end deffn @geindex next() (sage.databases.stein_watkins.SteinWatkinsAllData method) @anchor{sage/databases/stein_watkins sage databases stein_watkins SteinWatkinsAllData next}@anchor{3a} @deffn {Method} next () @end deffn @end deffn @geindex SteinWatkinsIsogenyClass (class in sage.databases.stein_watkins) @anchor{sage/databases/stein_watkins sage databases stein_watkins SteinWatkinsIsogenyClass}@anchor{3b} @deffn {Class} sage.databases.stein_watkins.SteinWatkinsIsogenyClass (conductor) @end deffn @geindex SteinWatkinsPrimeData (class in sage.databases.stein_watkins) @anchor{sage/databases/stein_watkins sage databases stein_watkins SteinWatkinsPrimeData}@anchor{3c} @deffn {Class} sage.databases.stein_watkins.SteinWatkinsPrimeData (num) Bases: @ref{38,,sage.databases.stein_watkins.SteinWatkinsAllData} @end deffn @geindex ecdb_num_curves() (in module sage.databases.stein_watkins) @anchor{sage/databases/stein_watkins sage databases stein_watkins ecdb_num_curves}@anchor{3d} @deffn {Function} sage.databases.stein_watkins.ecdb_num_curves (max_level=200000) Return a list whose @math{N}-th entry, for @code{0 <= N <= max_level}, is the number of elliptic curves of conductor @math{N} in the database. EXAMPLES: @example sage: sage.databases.stein_watkins.ecdb_num_curves(100) # optional - database_stein_watkins [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 6, 8, 0, 4, 0, 3, 4, 6, 0, 0, 6, 0, 5, 4, 0, 0, 8, 0, 4, 4, 4, 3, 4, 4, 5, 4, 4, 0, 6, 1, 2, 8, 2, 0, 6, 4, 8, 2, 2, 1, 6, 4, 6, 7, 3, 0, 0, 1, 4, 6, 4, 2, 12, 1, 0, 2, 4, 0, 6, 2, 0, 12, 1, 6, 4, 1, 8, 0, 2, 1, 6, 2, 0, 0, 1, 3, 16, 4, 3, 0, 2, 0, 8, 0, 6, 11, 4] @end example @end deffn @c nodoctest @node John Jones's tables of number fields,The On-Line Encyclopedia of Integer Sequences OEIS,The Stein-Watkins table of elliptic curves,Top @anchor{sage/databases/jones sage-databases-jones}@anchor{3e}@anchor{sage/databases/jones doc}@anchor{3f}@anchor{sage/databases/jones john-jones-s-tables-of-number-fields}@anchor{40} @chapter John Jones's tables of number fields @c This file has been autogenerated. @anchor{sage/databases/jones module-sage databases jones}@anchor{6} @geindex sage.databases.jones (module) In order to use the Jones database, the optional database package must be installed using the Sage command !sage -i database_jones_numfield This is a table of number fields with bounded ramification and degree @math{\leq 6}. You can query the database for all number fields in Jones's tables with bounded ramification and degree. EXAMPLES: First load the database: @example sage: J = JonesDatabase() sage: J John Jones's table of number fields with bounded ramification and degree <= 6 @end example List the degree and discriminant of all fields in the database that have ramification at most at 2: @example sage: [(k.degree(), k.disc()) for k in J.unramified_outside([2])] # optional - database_jones_numfield [(1, 1), (2, -4), (2, -8), (2, 8), (4, 256), (4, 512), (4, -1024), (4, -2048), (4, 2048), (4, 2048), (4, 2048)] @end example List the discriminants of the fields of degree exactly 2 unramified outside 2: @example sage: [k.disc() for k in J.unramified_outside([2],2)] # optional - database_jones_numfield [-4, -8, 8] @end example List the discriminants of cubic field in the database ramified exactly at 3 and 5: @example sage: [k.disc() for k in J.ramified_at([3,5],3)] # optional - database_jones_numfield [-135, -675, -6075, -6075] sage: factor(6075) 3^5 * 5^2 sage: factor(675) 3^3 * 5^2 sage: factor(135) 3^3 * 5 @end example List all fields in the database ramified at 101: @example sage: J.ramified_at(101) # optional - database_jones_numfield [Number Field in a with defining polynomial x^2 - 101, Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361, Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6, Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4, Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17] @end example @geindex JonesDatabase (class in sage.databases.jones) @anchor{sage/databases/jones sage databases jones JonesDatabase}@anchor{41} @deffn {Class} sage.databases.jones.JonesDatabase @geindex get() (sage.databases.jones.JonesDatabase method) @anchor{sage/databases/jones sage databases jones JonesDatabase get}@anchor{42} @deffn {Method} get (S, var='a') Return all fields in the database ramified exactly at the primes in S. INPUT: @itemize - @item @code{S} - list or set of primes, or a single prime @item @code{var} - the name used for the generator of the number fields (default 'a'). @end itemize EXAMPLES: @example sage: J = JonesDatabase() # optional - database_jones_numfield sage: J.get(163, var='z') # optional - database_jones_numfield [Number Field in z with defining polynomial x^2 + 163, Number Field in z with defining polynomial x^3 - x^2 - 54*x + 169, Number Field in z with defining polynomial x^4 - x^3 - 7*x^2 + 2*x + 9] sage: J.get([3, 4]) # optional - database_jones_numfield Traceback (most recent call last): ... ValueError: S must be a list of primes @end example @end deffn @geindex ramified_at() (sage.databases.jones.JonesDatabase method) @anchor{sage/databases/jones sage databases jones JonesDatabase ramified_at}@anchor{43} @deffn {Method} ramified_at (S, d=None, var='a') Return all fields in the database of degree d ramified exactly at the primes in S. The fields are ordered by degree and discriminant. INPUT: @itemize - @item @code{S} - list or set of primes @item @code{d} - None (default, in which case all fields of degree <= 6 are returned) or a positive integer giving the degree of the number fields returned. @item @code{var} - the name used for the generator of the number fields (default 'a'). @end itemize EXAMPLES: @example sage: J = JonesDatabase() # optional - database_jones_numfield sage: J.ramified_at([101,109]) # optional - database_jones_numfield [] sage: J.ramified_at([109]) # optional - database_jones_numfield [Number Field in a with defining polynomial x^2 - 109, Number Field in a with defining polynomial x^3 - x^2 - 36*x + 4, Number Field in a with defining polynomial x^4 - x^3 + 14*x^2 + 34*x + 393] sage: J.ramified_at(101) # optional - database_jones_numfield [Number Field in a with defining polynomial x^2 - 101, Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361, Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6, Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4, Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17] sage: J.ramified_at((2, 5, 29), 3, 'c') # optional - database_jones_numfield [Number Field in c with defining polynomial x^3 - x^2 - 8*x - 28, Number Field in c with defining polynomial x^3 - x^2 + 10*x + 102, Number Field in c with defining polynomial x^3 - x^2 + 97*x - 333, Number Field in c with defining polynomial x^3 - x^2 - 48*x - 188] @end example @end deffn @geindex unramified_outside() (sage.databases.jones.JonesDatabase method) @anchor{sage/databases/jones sage databases jones JonesDatabase unramified_outside}@anchor{44} @deffn {Method} unramified_outside (S, d=None, var='a') Return all fields in the database of degree d unramified outside S. If d is omitted, return fields of any degree up to 6. The fields are ordered by degree and discriminant. INPUT: @itemize - @item @code{S} - list or set of primes, or a single prime @item @code{d} - None (default, in which case all fields of degree <= 6 are returned) or a positive integer giving the degree of the number fields returned. @item @code{var} - the name used for the generator of the number fields (default 'a'). @end itemize EXAMPLES: @example sage: J = JonesDatabase() # optional - database_jones_numfield sage: J.unramified_outside([101,109]) # optional - database_jones_numfield [Number Field in a with defining polynomial x - 1, Number Field in a with defining polynomial x^2 - 101, Number Field in a with defining polynomial x^2 - 109, Number Field in a with defining polynomial x^3 - x^2 - 36*x + 4, Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361, Number Field in a with defining polynomial x^4 - x^3 + 14*x^2 + 34*x + 393, Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6, Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4, Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17] @end example @end deffn @end deffn @c nodoctest @node The On-Line Encyclopedia of Integer Sequences OEIS,Local copy of Sloane On-Line Encyclopedia of Integer Sequences,John Jones's tables of number fields,Top @anchor{sage/databases/oeis the-on-line-encyclopedia-of-integer-sequences-oeis}@anchor{45}@anchor{sage/databases/oeis doc}@anchor{46}@anchor{sage/databases/oeis sage-databases-oeis}@anchor{47} @chapter The On-Line Encyclopedia of Integer Sequences (OEIS) @c This file has been autogenerated. @anchor{sage/databases/oeis module-sage databases oeis}@anchor{8} @geindex sage.databases.oeis (module) You can query the OEIS (Online Database of Integer Sequences) through Sage in order to: @quotation @itemize - @item identify a sequence from its first terms. @item obtain more terms, formulae, references, etc. for a given sequence. @end itemize @end quotation AUTHORS: @itemize - @item Thierry Monteil (2012-02-10 -- 2013-06-21): initial version. @item Vincent Delecroix (2014): modifies continued fractions because of trac ticket #14567@footnote{https://trac.sagemath.org/14567} @item Moritz Firsching (2016): modifies handling of dead sequence, see trac ticket #17330@footnote{https://trac.sagemath.org/17330} @end itemize EXAMPLES: @example sage: oeis The On-Line Encyclopedia of Integer Sequences (http://oeis.org/) @end example What about a sequence starting with @math{3@comma{} 7@comma{} 15@comma{} 1} ? @example sage: search = oeis([3, 7, 15, 1], max_results=4) ; search # optional -- internet 0: A001203: Continued fraction expansion of Pi. 1: A082495: a(n) = (2^n - 1) mod n. 2: A165416: Irregular array read by rows: The n-th row contains those distinct positive integers that each, when written in binary, occurs as a substring in binary n. 3: A246674: Run Length Transform of A000225. sage: [u.id() for u in search] # optional -- internet ['A001203', 'A082495', 'A165416', 'A246674'] sage: c = search[0] ; c # optional -- internet A001203: Continued fraction expansion of Pi. @end example @example sage: c.first_terms(15) # optional -- internet (3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1) sage: c.examples() # optional -- internet 0: Pi = 3.1415926535897932384... 1: = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...)))) 2: = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...] sage: c.comments() # optional -- internet 0: The first 5,821,569,425 terms were computed by _Eric W. Weisstein_ on Sep 18 2011. 1: The first 10,672,905,501 terms were computed by _Eric W. Weisstein_ on Jul 17 2013. 2: The first 15,000,000,000 terms were computed by _Eric W. Weisstein_ on Jul 27 2013. @end example @example sage: x = c.natural_object() ; type(x) # optional -- internet sage: x.convergents()[:7] # optional -- internet [3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317] sage: RR(x.value()) # optional -- internet 3.14159265358979 sage: RR(x.value()) == RR(pi) # optional -- internet True @end example What about posets ? Are they hard to count ? To which other structures are they related ? @example sage: [Posets(i).cardinality() for i in range(10)] [1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231] sage: oeis(_) # optional -- internet 0: A000112: Number of partially ordered sets ("posets") with n unlabeled elements. sage: p = _[0] # optional -- internet @end example @example sage: 'hard' in p.keywords() # optional -- internet True sage: len(p.formulas()) # optional -- internet 0 sage: len(p.first_terms()) # optional -- internet 17 @end example @example sage: p.cross_references(fetch=True) # optional -- internet 0: A000798: Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements. 1: A001035: Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs). 2: A001930: Number of topologies, or transitive digraphs with n unlabeled nodes. 3: A006057: Number of topologies on n labeled points satisfying axioms T_0-T_4. 4: A079263: Number of constrained mixed models with n factors. 5: A079265: Number of antisymmetric transitive binary relations on n unlabeled points. @end example What does the Taylor expansion of the @math{e^(e^x-1)`} function have to do with primes ? @example sage: x = var('x') ; f(x) = e^(e^x - 1) sage: L = [a*factorial(b) for a,b in taylor(f(x), x, 0, 20).coefficients()] ; L [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372] sage: oeis(L) # optional -- internet 0: A000110: Bell or exponential numbers: number of ways to partition a set of n labeled elements. sage: b = _[0] # optional -- internet sage: b.formulas()[0] # optional -- internet 'E.g.f.: exp(exp(x) - 1).' sage: [i for i in b.comments() if 'prime' in i][-1] # optional -- internet 'Number n is prime if mod(a(n)-2,n) = 0. -_Dmitry Kruchinin_, Feb 14 2012' sage: [n for n in range(2, 20) if (b(n)-2) % n == 0] # optional -- internet [2, 3, 5, 7, 11, 13, 17, 19] @end example @subsubheading See also @itemize - @item If you plan to do a lot of automatic searches for subsequences, you should consider installing @ref{9,,SloaneEncyclopedia}, a local partial copy of the OEIS. @item Some infinite OEIS sequences are implemented in Sage, via the sloane_functions@footnote{../../../../../../html/en/reference/combinat/sage/combinat/sloane_functions.html#module-sage.combinat.sloane_functions} module. @end itemize @cartouche @quotation Todo @itemize - @item in case of flood, suggest the user to install the off-line database instead. @item interface with the off-line database (or reimplement it). @end itemize @end quotation @end cartouche @menu * Classes and methods:: @end menu @node Classes and methods,,,The On-Line Encyclopedia of Integer Sequences OEIS @anchor{sage/databases/oeis classes-and-methods}@anchor{48} @section Classes and methods @geindex FancyTuple (class in sage.databases.oeis) @anchor{sage/databases/oeis sage databases oeis FancyTuple}@anchor{49} @deffn {Class} sage.databases.oeis.FancyTuple Bases: tuple@footnote{https://docs.python.org/library/functions.html#tuple} This class inherits from @code{tuple}, it allows to nicely print tuples whose elements have a one line representation. EXAMPLES: @example sage: from sage.databases.oeis import FancyTuple sage: t = FancyTuple(['zero', 'one', 'two', 'three', 4]) ; t 0: zero 1: one 2: two 3: three 4: 4 sage: t[2] 'two' @end example @end deffn @geindex OEIS (class in sage.databases.oeis) @anchor{sage/databases/oeis sage databases oeis OEIS}@anchor{4a} @deffn {Class} sage.databases.oeis.OEIS The On-Line Encyclopedia of Integer Sequences. @code{OEIS} is a class representing the On-Line Encyclopedia of Integer Sequences. You can query it using its methods, but @code{OEIS} can also be called directly with three arguments: @itemize - @item @code{query} - it can be: @itemize - @item a string representing an OEIS ID (e.g. 'A000045'). @item an integer representing an OEIS ID (e.g. 45). @item a list representing a sequence of integers. @item a string, representing a text search. @end itemize @item @code{max_results} - (integer, default: 30) the maximum number of results to return, they are sorted according to their relevance. In any cases, the OEIS website will never provide more than 100 results. @item @code{first_result} - (integer, default: 0) allow to skip the @code{first_result} first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences. @end itemize OUTPUT: @itemize - @item if @code{query} is an integer or an OEIS ID (e.g. 'A000045'), returns the associated OEIS sequence. @item if @code{query} is a string, returns a tuple of OEIS sequences whose description corresponds to the query. Those sequences can be used without the need to fetch the database again. @item if @code{query} is a list of integers, returns a tuple of OEIS sequences containing it as a subsequence. Those sequences can be used without the need to fetch the database again. @end itemize EXAMPLES: @example sage: oeis The On-Line Encyclopedia of Integer Sequences (http://oeis.org/) @end example A particular sequence can be called by its A-number or number: @example sage: oeis('A000040') # optional -- internet A000040: The prime numbers. sage: oeis(45) # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. @end example The database can be searched by subsequence: @example sage: search = oeis([1,2,3,5,8,13]) ; search # optional -- internet 0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. 1: A027926: Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2. 2: A001129: Iccanobif numbers: reverse digits of two previous terms and add. sage: fibo = search[0] # optional -- internet sage: fibo.name() # optional -- internet 'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.' sage: fibo.first_terms() # optional -- internet (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169) sage: fibo.cross_references()[0] # optional -- internet 'A039834' sage: fibo == oeis(45) # optional -- internet True sage: sfibo = oeis('A039834') # optional -- internet sage: sfibo.first_terms() # optional -- internet (1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, 233, -377, 610, -987, 1597, -2584, 4181, -6765, 10946, -17711, 28657, -46368, 75025, -121393, 196418, -317811, 514229, -832040, 1346269, -2178309, 3524578, -5702887, 9227465, -14930352, 24157817) sage: sfibo.first_terms(absolute_value=True)[2:20] == fibo.first_terms()[:18] # optional -- internet True sage: fibo.formulas()[4] # optional -- internet 'F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).' sage: fibo.comments()[1] # optional -- internet "F(n+2) = number of binary sequences of length n that have no consecutive 0's." sage: fibo.links()[0] # optional -- internet 'http://oeis.org/A000045/b000045.txt' @end example The database can be searched by description: @example sage: oeis('prime gap factorization', max_results=4) # optional -- internet 0: A073491: Numbers having no prime gaps in their factorization. 1: A073490: Number of prime gaps in factorization of n. 2: A073492: Numbers having at least one prime gap in their factorization. 3: A073493: Numbers having exactly one prime gap in their factorization. @end example @cartouche @quotation Warning The following will fetch the OEIS database twice (once for searching the database, and once again for creating the sequence @code{fibo}): @example sage: oeis([1,2,3,5,8,13]) # optional -- internet 0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. 1: A027926: Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2. 2: A001129: Iccanobif numbers: reverse digits of two previous terms and add. sage: fibo = oeis('A000045') # optional -- internet @end example Do not do this, it is slow, it costs bandwidth and server resources ! Instead, do the following, to reuse the result of the search to create the sequence: @example sage: oeis([1,2,3,5,8,13]) # optional -- internet 0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. 1: A027926: Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2. 2: A001129: Iccanobif numbers: reverse digits of two previous terms and add. sage: fibo = _[0] # optional -- internet @end example @end quotation @end cartouche @geindex browse() (sage.databases.oeis.OEIS method) @anchor{sage/databases/oeis sage databases oeis OEIS browse}@anchor{4b} @deffn {Method} browse () Open the OEIS web page in a browser. EXAMPLES: @example sage: oeis.browse() # optional -- webbrowser @end example @end deffn @geindex find_by_description() (sage.databases.oeis.OEIS method) @anchor{sage/databases/oeis sage databases oeis OEIS find_by_description}@anchor{4c} @deffn {Method} find_by_description (description, max_results=3, first_result=0) Search for OEIS sequences corresponding to the description. INPUT: @itemize - @item @code{description} - (string) the description the searched sequences. @item @code{max_results} - (integer, default: 3) the maximum number of results we want. In any case, the on-line encyclopedia will not return more than 100 results. @item @code{first_result} - (integer, default: 0) allow to skip the @code{first_result} first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences. @end itemize OUTPUT: @itemize - @item a tuple (with fancy formatting) of at most @code{max_results} OEIS sequences. Those sequences can be used without the need to fetch the database again. @end itemize EXAMPLES: @example sage: oeis.find_by_description('prime gap factorization') # optional -- internet 0: A073491: Numbers having no prime gaps in their factorization. 1: A073490: Number of prime gaps in factorization of n. 2: A073492: Numbers having at least one prime gap in their factorization. sage: prime_gaps = _[1] ; prime_gaps # optional -- internet A073490: Number of prime gaps in factorization of n. @end example @example sage: oeis('beaver') # optional -- internet 0: A028444: Busy Beaver sequence, or Rado's sigma function: ... 1: A060843: Busy Beaver problem: a(n) = maximal number of steps ... 2: A131956: Busy Beaver variation: maximum number of steps for ... sage: oeis('beaver', max_results=4, first_result=2) # optional -- internet 0: A131956: Busy Beaver variation: maximum number of steps for ... 1: A141475: Number of Turing machines with n states following ... 2: A131957: Busy Beaver sigma variation: maximum number of 1's ... 3: A052200: Number of n-state, 2-symbol, d+ in @{LEFT, RIGHT@}, ... @end example @end deffn @geindex find_by_id() (sage.databases.oeis.OEIS method) @anchor{sage/databases/oeis sage databases oeis OEIS find_by_id}@anchor{4d} @deffn {Method} find_by_id (ident) INPUT: @itemize - @item @code{ident} - a string representing the A-number of the sequence or an integer representing its number. @end itemize OUTPUT: @itemize - @item The OEIS sequence whose A-number or number corresponds to @code{ident}. @end itemize EXAMPLES: @example sage: oeis.find_by_id('A000040') # optional -- internet A000040: The prime numbers. sage: oeis.find_by_id(40) # optional -- internet A000040: The prime numbers. @end example @end deffn @geindex find_by_subsequence() (sage.databases.oeis.OEIS method) @anchor{sage/databases/oeis sage databases oeis OEIS find_by_subsequence}@anchor{4e} @deffn {Method} find_by_subsequence (subsequence, max_results=3, first_result=0) Search for OEIS sequences containing the given subsequence. INPUT: @itemize - @item @code{subsequence} - a list of integers. @item @code{max_results} - (integer, default: 3), the maximum of results requested. @item @code{first_result} - (integer, default: 0) allow to skip the @code{first_result} first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences. @end itemize OUTPUT: @itemize - @item a tuple (with fancy formatting) of at most @code{max_results} OEIS sequences. Those sequences can be used without the need to fetch the database again. @end itemize EXAMPLES: @example sage: oeis.find_by_subsequence([2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]) # optional -- internet 0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. 1: A177194: Fibonacci numbers whose decimal expression does not contain any digit 0. 2: A212804: Expansion of (1-x)/(1-x-x^2). sage: fibo = _[0] ; fibo # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. @end example @end deffn @end deffn @geindex OEISSequence (class in sage.databases.oeis) @anchor{sage/databases/oeis sage databases oeis OEISSequence}@anchor{4f} @deffn {Class} sage.databases.oeis.OEISSequence (entry) Bases: sage.structure.sage_object.SageObject@footnote{../../../../../../html/en/reference/structure/sage/structure/sage_object.html#sage.structure.sage_object.SageObject} The class of OEIS sequences. This class implements OEIS sequences. Such sequences are produced from a string in the OEIS format. They are usually produced by calls to the On-Line Encyclopedia of Integer Sequences, represented by the class @ref{4a,,OEIS}. @cartouche @quotation Note Since some sequences do not start with index 0, there is a difference between calling and getting item, see @ref{50,,__call__()} for more details @example sage: sfibo = oeis('A039834') # optional -- internet sage: sfibo.first_terms()[:10] # optional -- internet (1, 1, 0, 1, -1, 2, -3, 5, -8, 13) sage: sfibo(-2) # optional -- internet 1 sage: sfibo(3) # optional -- internet 2 sage: sfibo.offsets() # optional -- internet (-2, 6) sage: sfibo[0] # optional -- internet 1 sage: sfibo[6] # optional -- internet -3 @end example @end quotation @end cartouche @geindex __call__() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence __call__}@anchor{50} @deffn {Method} __call__ (k) Return the element of the sequence @code{self} with index @code{k}. INPUT: @itemize - @item @code{k} - integer. @end itemize OUTPUT: @itemize - @item integer. @end itemize @cartouche @quotation Note The first index of the sequence @code{self} is not necessarily zero, it depends on the first offset of @code{self}. If the sequence represents the decimal expansion of a real number, the index 0 corresponds to the digit right after the decimal point. @end quotation @end cartouche EXAMPLES: @example sage: f = oeis(45) # optional -- internet sage: f.first_terms()[:10] # optional -- internet (0, 1, 1, 2, 3, 5, 8, 13, 21, 34) sage: f(4) # optional -- internet 3 @end example @example sage: sfibo = oeis('A039834') # optional -- internet sage: sfibo.first_terms()[:10] # optional -- internet (1, 1, 0, 1, -1, 2, -3, 5, -8, 13) sage: sfibo(-2) # optional -- internet 1 sage: sfibo(4) # optional -- internet -3 sage: sfibo.offsets() # optional -- internet (-2, 6) @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s(38) 1 sage: s(42) -1 sage: s(2) Traceback (most recent call last): ... ValueError: Sequence A999999 is not defined (or known) for index 2 @end example @end deffn @geindex author() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence author}@anchor{51} @deffn {Method} author () Returns the author of the sequence in the encyclopedia. OUTPUT: @itemize - @item string. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.author() # optional -- internet '_N. J. A. Sloane_, Apr 30 1991' @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.author() 'Anonymous.' @end example @end deffn @geindex browse() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence browse}@anchor{52} @deffn {Method} browse () Open the OEIS web page associated to the sequence @code{self} in a browser. EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet webbrowser A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.browse() # optional -- internet webbrowser @end example TESTS: @example sage: s = oeis._imaginary_sequence() # optional -- webbrowser sage: s.browse() # optional -- webbrowser @end example @end deffn @geindex comments() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence comments}@anchor{53} @deffn {Method} comments () Return a tuple of comments associated to the sequence @code{self}. OUTPUT: @itemize - @item tuple of strings (with fancy formatting). @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.comments()[:3] # optional -- internet 0: Also sometimes called Lamé's sequence. 1: F(n+2) = number of binary sequences of length n that have no consecutive 0's. 2: F(n+2) = number of subsets of @{1,2,...,n@} that contain no consecutive integers. @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.comments() 0: 42 is the product of the first 4 prime numbers, except 5 and perhaps 1. 1: Apart from that, i have no comment. @end example @end deffn @geindex cross_references() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence cross_references}@anchor{54} @deffn {Method} cross_references (fetch=False) Return a tuple of cross references associated to the sequence @code{self}. INPUT: @itemize - @item @code{fetch} - boolean (default: @code{False}). @end itemize OUTPUT: @itemize - @item if @code{fetch} is @code{False}, return a list of OEIS IDs (strings). @item if @code{fetch} if @code{True}, return a tuple of OEIS sequences. @end itemize EXAMPLES: @example sage: nbalanced = oeis("A005598") ; nbalanced # optional -- internet A005598: a(n)=1+sum((n-i+1)*phi(i),i=1..n). sage: nbalanced.cross_references() # optional -- internet ('A049703', 'A049695', 'A103116', 'A000010') sage: nbalanced.cross_references(fetch=True) # optional -- internet 0: A049703: a(0) = 0; for n>0, a(n) = A005598(n)/2. 1: A049695: Array T read by diagonals; T(i,j)=number of nonnegative slopes of lines determined by 2 lattice points in [ 0,i ] X [ 0,j ] if i>0; T(0,j)=1 if j>0; T(0,0)=0. 2: A103116: A005598(n) - 1. 3: A000010: Euler totient function phi(n): count numbers <= n and prime to n. sage: phi = _[3] # optional -- internet @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.cross_references() ('A000042', 'A000024') @end example @end deffn @geindex examples() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence examples}@anchor{55} @deffn {Method} examples () Return a tuple of examples associated to the sequence @code{self}. OUTPUT: @itemize - @item tuple of strings (with fancy formatting). @end itemize EXAMPLES: @example sage: c = oeis(1203) ; c # optional -- internet A001203: Continued fraction expansion of Pi. sage: c.examples() # optional -- internet 0: Pi = 3.1415926535897932384... 1: = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...)))) 2: = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...] @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.examples() 0: s(42) + s(43) = 0. @end example @end deffn @geindex extensions_or_errors() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence extensions_or_errors}@anchor{56} @deffn {Method} extensions_or_errors () Return a tuple of extensions or errors associated to the sequence @code{self}. OUTPUT: @itemize - @item tuple of strings (with fancy formatting). @end itemize EXAMPLES: @example sage: sfibo = oeis('A039834') ; sfibo # optional -- internet A039834: a(n+2) = -a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers (A000045) extended to negative indices. sage: sfibo.extensions_or_errors()[0] # optional -- internet 'Signs corrected by _Len Smiley_ and _N. J. A. Sloane_.' @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.extensions_or_errors() 0: This sequence does not contain errors. @end example @end deffn @geindex first_terms() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence first_terms}@anchor{57} @deffn {Method} first_terms (number=None, absolute_value=False) INPUT: @itemize - @item @code{number} - (integer or @code{None}, default: @code{None}) the number of terms returned (if less than the number of available terms). When set to None, returns all the known terms. @item @code{absolute_value} - (bool, default: @code{False}) when a sequence has negative entries, OEIS also stores the absolute values of its first terms, when @code{absolute_value} is set to @code{True}, you will get them. @end itemize OUTPUT: @itemize - @item tuple of integers. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.first_terms()[:10] # optional -- internet (0, 1, 1, 2, 3, 5, 8, 13, 21, 34) @end example Handle dead sequences: see trac ticket #17330@footnote{https://trac.sagemath.org/17330} @example sage: oeis(17).first_terms(12) # optional -- internet oeis(17).first_terms(12) .. RuntimeWarning: This sequence is dead "A000017: Erroneous version of A032522." warn('This sequence is dead "'+self.id()+": "+self.name()+'"', RuntimeWarning) (1, 0, 0, 2, 2, 4, 8, 4, 16, 12, 48, 80) @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.first_terms() (1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) sage: s.first_terms(5) (1, 1, 1, 1, -1) sage: s.first_terms(5, absolute_value=True) (1, 1, 1, 1, 1) sage: s = oeis._imaginary_sequence(keywords='full') sage: s(40) Traceback (most recent call last): ... TypeError: You found a sign inconsistency, please contact OEIS sage: s = oeis._imaginary_sequence(keywords='sign,full') sage: s(40) 1 sage: s = oeis._imaginary_sequence(keywords='nonn,full') sage: s(42) 1 @end example @end deffn @geindex formulas() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence formulas}@anchor{58} @deffn {Method} formulas () Return a tuple of formulas associated to the sequence @code{self}. OUTPUT: @itemize - @item tuple of strings (with fancy formatting). @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.formulas()[2] # optional -- internet 'F(n) = ((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5)).' @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.formulas() 0: For n big enough, s(n+1) - s(n) = 0. @end example @end deffn @geindex id() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence id}@anchor{59} @deffn {Method} id (format='A') The ID of the sequence @code{self} is the A-number that identifies @code{self}. INPUT: @itemize - @item @code{format} - (string, default: 'A'). @end itemize OUTPUT: @itemize - @item if @code{format} is set to 'A', returns a string of the form 'A000123'. @item if @code{format} is set to 'int' returns an integer of the form 123. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.id() # optional -- internet 'A000045' sage: f.id(format='int') # optional -- internet 45 @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.id() 'A999999' sage: s.id(format='int') 999999 @end example @end deffn @geindex is_finite() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence is_finite}@anchor{5a} @deffn {Method} is_finite () Tells whether the sequence is finite. Currently, OEIS only provides a keyword when the sequence is known to be finite. So, when this keyword is not there, we do not know whether it is infinite or not. OUTPUT: @itemize - @item Returns @code{True} when the sequence is known to be finite. @item Returns @code{Unknown} otherwise. @end itemize @cartouche @quotation Todo Ask OEIS for a keyword ensuring that a sequence is infinite. @end quotation @end cartouche EXAMPLES: @example sage: s = oeis('A114288') ; s # optional -- internet A114288: Lexicographically earliest solution of any 9 X 9 sudoku, read by rows. sage: s.is_finite() # optional -- internet True @end example @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.is_finite() # optional -- internet Unknown @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.is_finite() Unknown sage: s = oeis._imaginary_sequence('nonn,finit') sage: s.is_finite() True @end example @end deffn @geindex is_full() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence is_full}@anchor{5b} @deffn {Method} is_full () Tells whether the sequence @code{self} is full, that is, if all its elements are listed in @code{self.first_terms()}. Currently, OEIS only provides a keyword when the sequence is known to be full. So, when this keyword is not there, we do not know whether some elements are missing or not. OUTPUT: @itemize - @item Returns @code{True} when the sequence is known to be full. @item Returns @code{Unknown} otherwise. @end itemize EXAMPLES: @example sage: s = oeis('A114288') ; s # optional -- internet A114288: Lexicographically earliest solution of any 9 X 9 sudoku, read by rows. sage: s.is_full() # optional -- internet True @end example @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.is_full() # optional -- internet Unknown @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.is_full() Unknown sage: s = oeis._imaginary_sequence('nonn,full,finit') sage: s.is_full() True @end example @end deffn @geindex keywords() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence keywords}@anchor{5c} @deffn {Method} keywords () Return the keywords associated to the sequence @code{self}. OUTPUT: @itemize - @item tuple of strings. @end itemize EXAMPLES: @example sage: f = oeis(53) ; f # optional -- internet A000053: Local stops on New York City Broadway line (IRT #1) subway. sage: f.keywords() # optional -- internet ('nonn', 'fini', 'full') @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.keywords() ('sign', 'easy') sage: s = oeis._imaginary_sequence(keywords='nonn,hard') sage: s.keywords() ('nonn', 'hard') @end example @end deffn @geindex links() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence links}@anchor{5d} @deffn {Method} links (browse=None, format='guess') Return, display or browse links associated to the sequence @code{self}. INPUT: @itemize - @item @code{browse} - an integer, a list of integers, or the word 'all' (default: @code{None}) : which links to open in a web browser. @item @code{format} - string (default: 'guess') : how to display the links. @end itemize OUTPUT: @itemize - @item @table @asis @item tuple of strings (with fancy formatting): @itemize - @item if @code{format} is @code{url}, returns a tuple of absolute links without description. @item if @code{format} is @code{html}, returns nothing but prints a tuple of clickable absolute links in their context. @item if @code{format} is @code{guess}, adapts the output to the context (command line or notebook). @item if @code{format} is @code{raw}, the links as they appear in the database, relative links are not made absolute. @end itemize @end table @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.links(format='url') # optional -- internet 0: http://oeis.org/A000045/b000045.txt 1: http://www.schoolnet.ca/vp-pv/amof/e_fiboI.htm ... sage: f.links(format='raw') # optional -- internet 0: N. J. A. Sloane, The first 2000 Fibonacci numbers: Table of n, F(n) for n = 0..2000 1: Amazing Mathematical Object Factory, Information on the Fibonacci sequences ... @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.links(format='raw')[2] 'Do not confuse with the sequence A000042 or the sequence A000024' sage: s.links(format='url')[3] 'http://oeis.org/A000024' sage: HTML = s.links(format="html"); HTML 0: Wikipedia, 42 (number) 1: See. also trac ticket #42 ... sage: type(HTML) @end example @end deffn @geindex name() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence name}@anchor{5e} @deffn {Method} name () Return the name of the sequence @code{self}. OUTPUT: @itemize - @item string. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.name() # optional -- internet 'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.' @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.name() 'The opposite of twice the characteristic sequence of 42 plus one, starting from 38.' @end example @end deffn @geindex natural_object() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence natural_object}@anchor{5f} @deffn {Method} natural_object () Return the natural object associated to the sequence @code{self}. OUTPUT: @itemize - @item @table @asis @item If the sequence @code{self} corresponds to the digits of a real number, returns the associated real number (as an element of RealLazyField()). @end table @item @table @asis @item If the sequence @code{self} corresponds to the convergents of a continued fraction, returns the associated continued fraction. @end table @end itemize @cartouche @quotation Warning This method forgets the fact that the returned sequence may not be complete. @end quotation @end cartouche @cartouche @quotation Todo @itemize - @item ask OEIS to add a keyword telling whether the sequence comes from a power series, e.g. for @indicateurl{http://oeis.org/A000182} @item discover other possible conversions. @end itemize @end quotation @end cartouche EXAMPLES: @example sage: g = oeis("A002852") ; g # optional -- internet A002852: Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma. sage: x = g.natural_object() ; type(x) # optional -- internet sage: RDF(x) == RDF(euler_gamma) # optional -- internet True sage: cfg = continued_fraction(euler_gamma) sage: x[:90] == cfg[:90] # optional -- internet True @end example @example sage: ee = oeis('A001113') ; ee # optional -- internet A001113: Decimal expansion of e. sage: x = ee.natural_object() ; x # optional -- internet 2.718281828459046? sage: x.parent() # optional -- internet Real Lazy Field sage: x == RR(e) # optional -- internet True @end example @example sage: av = oeis('A087778') ; av # optional -- internet A087778: Decimal expansion of Avogadro's constant. sage: av.natural_object() # optional -- internet 6.022141000000000?e23 @end example @example sage: fib = oeis('A000045') ; fib # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: x = fib.natural_object() ; x.universe() # optional -- internet Non negative integer semiring @end example @example sage: sfib = oeis('A039834') ; sfib # optional -- internet A039834: a(n+2) = -a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers (A000045) extended to negative indices. sage: x = sfib.natural_object() ; x.universe() # optional -- internet Integer Ring @end example TESTS: @example sage: s = oeis._imaginary_sequence('nonn,cofr') sage: type(s.natural_object()) sage: s = oeis._imaginary_sequence('nonn') sage: s.natural_object().universe() Non negative integer semiring sage: s = oeis._imaginary_sequence() sage: s.natural_object().universe() Integer Ring @end example @end deffn @geindex offsets() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence offsets}@anchor{60} @deffn {Method} offsets () Return the offsets of the sequence @code{self}. The first offset is the subscript of the first term in the sequence @code{self}. When, the sequence represents the decimal expansion of a real number, it corresponds to the number of digits of its integer part. The second offset is the first term in the sequence @code{self} (starting from 1) whose absolute value is greater than 1. This is set to 1 if all the terms are 0 or +-1. OUTPUT: @itemize - @item tuple of two elements. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.offsets() # optional -- internet (0, 4) sage: f.first_terms()[:4] # optional -- internet (0, 1, 1, 2) @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.offsets() (38, 4) @end example @end deffn @geindex old_IDs() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence old_IDs}@anchor{61} @deffn {Method} old_IDs () Returns the IDs of the sequence @code{self} corresponding to ancestors of OEIS. OUTPUT: @itemize - @item a tuple of at most two strings. When the string starts with @math{M}, it corresponds to the ID of "The Encyclopedia of Integer Sequences" of 1995. When the string starts with @math{N}, it corresponds to the ID of the "Handbook of Integer Sequences" of 1973. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.old_IDs() # optional -- internet ('M0692', 'N0256') @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.old_IDs() ('M9999', 'N9999') @end example @end deffn @geindex programs() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence programs}@anchor{62} @deffn {Method} programs (language='other') Returns programs implementing the sequence @code{self} in the given @code{language}. INPUT: @itemize - @item @code{language} - string (default: 'other') - the language of the program. Current values are: 'maple', 'mathematica' and 'other'. @end itemize OUTPUT: @itemize - @item tuple of strings (with fancy formatting). @end itemize @cartouche @quotation Todo ask OEIS to add a "Sage program" field in the database ;) @end quotation @end cartouche EXAMPLES: @example sage: ee = oeis('A001113') ; ee # optional -- internet A001113: Decimal expansion of e. sage: ee.programs()[0] # optional -- internet '(PARI) @{ default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); @} \\\\ _Harry J. Smith_, Apr 15 2009' @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.programs() 0: (Python) 1: def A999999(n): 2: assert(isinstance(n, (int, Integer))), "n must be an integer." 3: if n < 38: 4: raise ValueError("The value %s is not accepted." %str(n))) 5: elif n == 42: 6: return -1 7: else: 8: return 1 sage: s.programs('maple') 0: Do not even try, Maple is not able to produce such a sequence. sage: s.programs('mathematica') 0: Mathematica neither. @end example @end deffn @geindex raw_entry() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence raw_entry}@anchor{63} @deffn {Method} raw_entry () Return the raw entry of the sequence @code{self}, in the OEIS format. OUTPUT: @itemize - @item string. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: print(f.raw_entry()) # optional -- internet %I A000045 M0692 N0256 %S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,... %T A000045 10946,17711,28657,46368,... ... @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.raw_entry() == oeis._imaginary_entry('sign,easy') True @end example @end deffn @geindex references() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence references}@anchor{64} @deffn {Method} references () Return a tuple of references associated to the sequence @code{self}. OUTPUT: @itemize - @item tuple of strings (with fancy formatting). @end itemize EXAMPLES: @example sage: w = oeis(7540) ; w # optional -- internet A007540: Wilson primes: primes p such that (p-1)! == -1 (mod p^2). sage: w.references() # optional -- internet 0: A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52. 1: C. Clawson, Mathematical Mysteries, Plenum Press, 1996, p. 180. 2: R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29. 3: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 80. ... sage: _[0] # optional -- internet 'A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.' @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.references()[1] 'Lewis Carroll, The Hunting of the Snark.' @end example @end deffn @geindex show() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence show}@anchor{65} @deffn {Method} show () Display most available informations about the sequence @code{self}. EXAMPLES: @example sage: s = oeis(12345) # optional -- internet sage: s.show() # optional -- internet ID A012345 NAME Coefficients in the expansion sinh(arcsin(x)*arcsin(x)) = 2*x^2/2!+8*x^4/4!+248*x^6/6!+11328*x^8/8!+... FIRST TERMS (2, 8, 248, 11328, 849312, 94857600, 14819214720, 3091936512000, 831657655349760, 280473756197529600, 115967597965430077440, 57712257892456911912960, 34039765801079493369569280) FORMULAS ... OFFSETS (0, 1) URL http://oeis.org/A012345 AUTHOR Patrick Demichel (patrick.demichel(AT)hp.com) @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.show() ID A999999 NAME The opposite of twice the characteristic sequence of 42 plus ... FIRST TERMS (1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... COMMENTS 0: 42 is the product of the first 4 prime numbers, except ... 1: Apart from that, i have no comment. ... @end example @end deffn @geindex url() (sage.databases.oeis.OEISSequence method) @anchor{sage/databases/oeis sage databases oeis OEISSequence url}@anchor{66} @deffn {Method} url () Return the URL of the page associated to the sequence @code{self}. OUTPUT: @itemize - @item string. @end itemize EXAMPLES: @example sage: f = oeis(45) ; f # optional -- internet A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. sage: f.url() # optional -- internet 'http://oeis.org/A000045' @end example TESTS: @example sage: s = oeis._imaginary_sequence() sage: s.url() 'http://oeis.org/A999999' @end example @end deffn @end deffn @geindex to_tuple() (in module sage.databases.oeis) @anchor{sage/databases/oeis sage databases oeis to_tuple}@anchor{67} @deffn {Function} sage.databases.oeis.to_tuple (string) @end deffn @c nodoctest @node Local copy of Sloane On-Line Encyclopedia of Integer Sequences,FindStat - the Combinatorial Statistic Finder,The On-Line Encyclopedia of Integer Sequences OEIS,Top @anchor{sage/databases/sloane sage-databases-sloane}@anchor{68}@anchor{sage/databases/sloane doc}@anchor{69}@anchor{sage/databases/sloane local-copy-of-sloane-on-line-encyclopedia-of-integer-sequences}@anchor{6a} @chapter Local copy of Sloane On-Line Encyclopedia of Integer Sequences @c This file has been autogenerated. @anchor{sage/databases/sloane module-sage databases sloane}@anchor{9} @geindex sage.databases.sloane (module) The SloaneEncyclopedia object provides access to a local copy of the database containing only the sequences and their names. To use this you must download and install the database using @code{SloaneEncyclopedia.install()}, or @code{SloaneEncyclopedia.install_from_gz()} if you have already downloaded the database manually. To look up a sequence, type @example sage: SloaneEncyclopedia[60843] # optional - sloane_database [1, 6, 21, 107] @end example To get the name of a sequence, type @example sage: SloaneEncyclopedia.sequence_name(1) # optional - sloane_database 'Number of groups of order n.' @end example To search locally for a particular subsequence, type @example sage: SloaneEncyclopedia.find([1,2,3,4,5], 1) # optional - sloane_database [(15, [1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13, 13, 16, 16, 16, 17, 19, 19, 23, 23, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 32, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 49, 49, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 64, 64, 64, 67, 67, 67, 71, 71, 71, 71, 73])] @end example The default maximum number of results is 30, but to return up to 100, type @example sage: SloaneEncyclopedia.find([1,2,3,4,5], 100) # optional - sloane_database [(15, [1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, ... @end example Results in either case are of the form [ (number, list) ]. @subsubheading See also @itemize - @item If you want to get more informations relative to a sequence (references, links, examples, programs, ...), you can use the On-Line Encyclopedia of Integer Sequences provided by the @ref{8,,OEIS} module. @item Some infinite OEIS sequences are implemented in Sage, via the sloane_functions@footnote{../../../../../../html/en/reference/combinat/sage/combinat/sloane_functions.html#module-sage.combinat.sloane_functions} module. @end itemize AUTHORS: @itemize - @item Steven Sivek (2005-12-22): first version @item Steven Sivek (2006-02-07): updated to correctly handle the new search form on the Sloane website, and it's now also smarter about loading the local database in that it doesn't convert a sequence from string form to a list of integers until absolutely necessary. This seems to cut the loading time roughly in half. @item Steven Sivek (2009-12-22): added the SloaneEncyclopedia functions install() and install_from_gz() so users can get the latest versions of the OEIS without having to get an updated spkg; added sequence_name() to return the description of a sequence; and changed the data type for elements of each sequence from int to Integer. @item Thierry Monteil (2012-02-10): deprecate dead code and update related doc and tests. @end itemize @menu * Classes and methods: Classes and methods<2>. @end menu @node Classes and methods<2>,,,Local copy of Sloane On-Line Encyclopedia of Integer Sequences @anchor{sage/databases/sloane classes-and-methods}@anchor{6b} @section Classes and methods @geindex SloaneEncyclopediaClass (class in sage.databases.sloane) @anchor{sage/databases/sloane sage databases sloane SloaneEncyclopediaClass}@anchor{6c} @deffn {Class} sage.databases.sloane.SloaneEncyclopediaClass A local copy of the Sloane Online Encyclopedia of Integer Sequences that contains only the sequence numbers and the sequences themselves. @geindex find() (sage.databases.sloane.SloaneEncyclopediaClass method) @anchor{sage/databases/sloane sage databases sloane SloaneEncyclopediaClass find}@anchor{6d} @deffn {Method} find (seq, maxresults=30) Return a list of all sequences which have seq as a subsequence, up to maxresults results. Sequences are returned in the form (number, list). INPUT: @itemize - @item @code{seq} - list @item @code{maxresults} - int @end itemize OUTPUT: list of 2-tuples (i, v), where v is a sequence with seq as a subsequence. @end deffn @geindex install() (sage.databases.sloane.SloaneEncyclopediaClass method) @anchor{sage/databases/sloane sage databases sloane SloaneEncyclopediaClass install}@anchor{6e} @deffn {Method} install (oeis_url='http://oeis.org/stripped.gz', names_url='http://oeis.org/names.gz', overwrite=False) Download and install the online encyclopedia, raising an IOError if either step fails. INPUT: @itemize - @item @code{oeis_url} - string (default: "@indicateurl{http://oeis.org}...") The URL of the stripped.gz encyclopedia file. @item @code{names_url} - string (default: "@indicateurl{http://oeis.org}...") The URL of the names.gz encyclopedia file. If you do not want to download this file, set names_url=None. @item @code{overwrite} - boolean (default: False) If the encyclopedia is already installed and overwrite=True, download and install the latest version over the installed one. @end itemize @end deffn @geindex install_from_gz() (sage.databases.sloane.SloaneEncyclopediaClass method) @anchor{sage/databases/sloane sage databases sloane SloaneEncyclopediaClass install_from_gz}@anchor{6f} @deffn {Method} install_from_gz (stripped_file, names_file, overwrite=False) Install the online encyclopedia from a local stripped.gz file. INPUT: @itemize - @item @code{stripped_file} - string. The name of the stripped.gz OEIS file. @item @code{names_file} - string. The name of the names.gz OEIS file, or None if the user does not want it installed. @item @code{overwrite} - boolean (default: False) If the encyclopedia is already installed and overwrite=True, install 'filename' over the old encyclopedia. @end itemize @end deffn @geindex load() (sage.databases.sloane.SloaneEncyclopediaClass method) @anchor{sage/databases/sloane sage databases sloane SloaneEncyclopediaClass load}@anchor{70} @deffn {Method} load () Load the entire encyclopedia into memory from a file. This is done automatically if the user tries to perform a lookup or a search. @end deffn @geindex sequence_name() (sage.databases.sloane.SloaneEncyclopediaClass method) @anchor{sage/databases/sloane sage databases sloane SloaneEncyclopediaClass sequence_name}@anchor{71} @deffn {Method} sequence_name (N) Return the name of sequence N in the encyclopedia. If sequence N does not exist, return '@w{'}. If the names database is not installed, raise an IOError. INPUT: @itemize - @item @code{N} - int @end itemize OUTPUT: string EXAMPLES: sage: SloaneEncyclopedia.sequence_name(1) # optional - sloane_database 'Number of groups of order n.' @end deffn @geindex unload() (sage.databases.sloane.SloaneEncyclopediaClass method) @anchor{sage/databases/sloane sage databases sloane SloaneEncyclopediaClass unload}@anchor{72} @deffn {Method} unload () Remove the database from memory. @end deffn @end deffn @geindex copy_gz_file() (in module sage.databases.sloane) @anchor{sage/databases/sloane sage databases sloane copy_gz_file}@anchor{73} @deffn {Function} sage.databases.sloane.copy_gz_file (gz_source, bz_destination) Decompress a gzipped file and install the bzipped verson. This is used by SloaneEncyclopedia.install_from_gz to install several gzipped OEIS database files. INPUT: @itemize - @item @code{gz_source} - string. The name of the gzipped file. @item @code{bz_destination} - string. The name of the newly compressed file. @end itemize @end deffn @geindex parse_sequence() (in module sage.databases.sloane) @anchor{sage/databases/sloane sage databases sloane parse_sequence}@anchor{74} @deffn {Function} sage.databases.sloane.parse_sequence (text='@w{'}) This internal function was only used by the sloane_find function, which is now deprecated. TESTS: @example sage: from sage.databases.sloane import parse_sequence sage: parse_sequence() doctest:...: DeprecationWarning: The function parse_sequence is not used anymore (2012-01-01). See http://trac.sagemath.org/10358 for details. @end example @end deffn @geindex sloane_find() (in module sage.databases.sloane) @anchor{sage/databases/sloane sage databases sloane sloane_find}@anchor{75} @deffn {Function} sage.databases.sloane.sloane_find (list=[], nresults=30, verbose=True) This function is broken. It is replaced by the @ref{8,,OEIS} module. Type @code{oeis?} for more information. TESTS: @example sage: sloane_find([1,2,3]) doctest:...: DeprecationWarning: The function sloane_find is deprecated. Use oeis() instead (2012-01-01). See http://trac.sagemath.org/10358 for details. @end example @end deffn @geindex sloane_sequence() (in module sage.databases.sloane) @anchor{sage/databases/sloane sage databases sloane sloane_sequence}@anchor{76} @deffn {Function} sage.databases.sloane.sloane_sequence (number=1, verbose=True) This function is broken. It is replaced by the @ref{8,,OEIS} module. Type @code{oeis?} for more information. TESTS: @example sage: sloane_sequence(123) doctest:...: DeprecationWarning: The function sloane_sequence is deprecated. Use oeis() instead (2012-01-01). See http://trac.sagemath.org/10358 for details. @end example @end deffn @c nodoctest @node FindStat - the Combinatorial Statistic Finder,Frank Luebeck's tables of Conway polynomials over finite fields,Local copy of Sloane On-Line Encyclopedia of Integer Sequences,Top @anchor{sage/databases/findstat sage-databases-findstat}@anchor{77}@anchor{sage/databases/findstat doc}@anchor{78}@anchor{sage/databases/findstat findstat-the-combinatorial-statistic-finder}@anchor{79} @chapter FindStat - the Combinatorial Statistic Finder. @c This file has been autogenerated. @anchor{sage/databases/findstat module-sage databases findstat}@anchor{5} @geindex sage.databases.findstat (module) The FindStat database can be found at @indicateurl{http://www.findstat.org} . Fix the following three notions: @itemize - @item A @emph{combinatorial collection} is a set @math{S} with interesting combinatorial properties, @item a @emph{combinatorial map} is a combinatorially interesting map @math{f: S \to S'} between combinatorial collections, and @item a @emph{combinatorial statistic} is a combinatorially interesting map @math{s: S \to \ZZ}. @end itemize You can use the sage interface to FindStat to: @itemize - @item identify a combinatorial statistic from the values on a few small objects, @item obtain more terms, formulae, references, etc. for a given statistic, @item edit statistics and submit new statistics. @end itemize To access the database, use @ref{7a,,findstat}: @example sage: findstat The Combinatorial Statistic Finder (http://www.findstat.org/) @end example AUTHORS: @itemize - @item Martin Rubey (2015): initial version. @end itemize @menu * A guided tour:: * Classes and methods: Classes and methods<3>. @end menu @node A guided tour,Classes and methods<3>,,FindStat - the Combinatorial Statistic Finder @anchor{sage/databases/findstat a-guided-tour}@anchor{7b} @section A guided tour @menu * Retrieving information:: * Editing and submitting statistics:: @end menu @node Retrieving information,Editing and submitting statistics,,A guided tour @anchor{sage/databases/findstat retrieving-information}@anchor{7c} @subsection Retrieving information The most straightforward application of the FindStat interface is to gather information about a combinatorial statistic. To do this, we supply @ref{7a,,findstat} with a list of @math{(object@comma{} value)} pairs. For example: @example sage: PM8 = PerfectMatchings(8) sage: r = findstat([(m, m.number_of_nestings()) for m in PM8]); r # optional -- internet,random 0: (St000041: The number of nestings of a perfect matching. , [], 105) ... @end example The result of this query is a list (presented as a @ref{49,,sage.databases.oeis.FancyTuple}) of triples. The first element of each triple is a @ref{7d,,FindStatStatistic} @math{s: S \to \ZZ}, the second element a list of @ref{7e,,FindStatMap}'s @math{f_i: S_i \to S_@{i+1@}}, and the third element is an integer: @example sage: (s, list_f, quality) = r[0] # optional -- internet @end example The precise meaning of the result is as follows: @quotation The composition @math{f_n \circ ... \circ f_2 \circ f_1} applied to the objects sent to FindStat agrees with @math{quality} many @math{(object@comma{} value)} pairs of @math{s} in the database. Moreover, there are no other @math{(object@comma{} value)} pairs of @math{s} stored in the database, i.e., there is no disagreement of values. @end quotation Put differently, if @math{quality} is not too small it is likely that the statistic sent to FindStat equals @math{s \circ f_n \circ ... \circ f_2 \circ f_1}. In the case at hand, the list of maps is empty and the integer @math{quality} equals the number of @math{(object@comma{} value)} pairs passed to FindStat. This means, that the set of @math{(object@comma{} value)} pairs of the statistic @math{s} as stored in the FindStat database is a superset of the data sent. We can now retrieve the description from the database: @example sage: print(s.description()) # optional -- internet,random The number of nestings of a perfect matching. This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$. @end example and check the references: @example sage: s.references() # optional -- internet,random 0: [1] [[MathSciNet:1288802]] 1: [2] [[MathSciNet:1418763]] @end example If you prefer, you can look at this information also in your browser: @example sage: findstat(41).browse() # optional -- webbrowser @end example Another interesting possibility is to look for equidistributed statistics. Instead of submitting a list of pairs, we pass a pair of lists: @example sage: r = findstat((PM8, [m.number_of_nestings() for m in PM8])); r # optional -- internet,random 0: (St000041: The number of nestings of a perfect matching. , [], 105) 1: (St000042: The number of crossings of a perfect matching. , [], 105) ... @end example This results tells us that the database contains another entriy that is equidistributed with the number of nestings on perfect matchings of length @math{8}, namely the number of crossings. Let us now look at a slightly more complicated example, where the submitted statistic is the composition of a sequence of combinatorial maps and a statistic known to FindStat. We use the occasion to advertise yet another way to pass values to FindStat: @example sage: r = findstat(Permutations(4), lambda pi: pi.saliances()[0]); r # optional -- internet,random 0: (St000051: The size of the left subtree. , [Mp00069: complement, Mp00061: to increasing tree], 24) ... sage: (s, list_f, quality) = r[0] # optional -- internet @end example To obtain the value of the statistic sent to FindStat on a given object, apply the maps in the list in the given order to this object, and evaluate the statistic on the result. For example, let us check that the result given by FindStat agrees with our statistic on the following permutation: @example sage: pi = Permutation([3,1,4,5,2]); pi.saliances()[0] 3 @end example We first have to find out, what the maps and the statistic actually do: @example sage: print(s.description()) # optional -- internet,random The size of the left subtree. sage: print(s.code()) # optional -- internet,random def statistic(T): return T[0].node_number() sage: print(list_f[0].code() + "\r\n" + list_f[1].code()) # optional -- internet,random def complement(elt): n = len(elt) return elt.__class__(elt.parent(), map(lambda x: n - x + 1, elt) ) def increasing_tree_shape(elt, compare=min): return elt.increasing_tree(compare).shape() @end example So, the following should coincide with what we sent FindStat: @example sage: pi.complement().increasing_tree_shape()[0].node_number() 3 @end example @node Editing and submitting statistics,,Retrieving information,A guided tour @anchor{sage/databases/findstat editing-and-submitting-statistics}@anchor{7f} @subsection Editing and submitting statistics Of course, often a statistic will not be in the database: @example sage: findstat([(d, randint(1,1000)) for d in DyckWords(4)]) # optional -- internet a new statistic on Cc0005: Dyck paths @end example In this case, and if the statistic might be "interesting", please consider submitting it to the database using @ref{80,,FindStatStatistic.submit()}. Also, you may notice omissions, typos or even mistakes in the description, the code and the references. In this case, simply replace the value by using @ref{81,,FindStatStatistic.set_description()}, @ref{82,,FindStatStatistic.set_code()} or @ref{83,,FindStatStatistic.set_references()}, and then @ref{80,,FindStatStatistic.submit()} your changes for review by the FindStat team. @node Classes and methods<3>,,A guided tour,FindStat - the Combinatorial Statistic Finder @anchor{sage/databases/findstat classes-and-methods}@anchor{84} @section Classes and methods @geindex FindStat (class in sage.databases.findstat) @anchor{sage/databases/findstat sage databases findstat FindStat}@anchor{7a} @deffn {Class} sage.databases.findstat.FindStat Bases: sage.structure.sage_object.SageObject@footnote{../../../../../../html/en/reference/structure/sage/structure/sage_object.html#sage.structure.sage_object.SageObject} The Combinatorial Statistic Finder. @ref{7a,,FindStat} is a class representing results of queries to the FindStat database. This class is also the entry point to edit statistics and new submissions. Use the shorthand @ref{7a,,findstat} to call it. INPUT: One of the following: @itemize - @item an integer or a string representing a valid FindStat identifier (e.g. 45 or 'St000045'). The keyword arguments @code{depth} and @code{max_values} are ignored. @item a list of pairs of the form (object, value), or a dictionary from sage objects to integer values. The keyword arguments @code{depth} and @code{max_values} are passed to the finder. @item a list of pairs of the form (list of objects, list of values), or a single pair of the form (list of objects, list of values). In each pair there should be as many objects as values. The keyword arguments @code{depth} and @code{max_values} are passed to the finder. @item a collection and a list of pairs of the form (string, value), or a dictionary from strings to integer values. The keyword arguments @code{depth} and @code{max_values} are passed to the finder. This should only be used if the collection is not yet supported. @item a collection and a list of pairs of the form (list of strings, list of values), or a single pair of the form (list of strings, list of values). In each pair there should be as many strings as values. The keyword arguments @code{depth} and @code{max_values} are passed to the finder. This should only be used if the collection is not yet supported. @item a collection and a callable. The callable is used to generate @code{max_values} (object, value) pairs. The number of terms generated may also be controlled by passing an iterable collection, such as @code{Permutations(3)}. The keyword arguments @code{depth} and @code{max_values} are passed to the finder. @end itemize OUTPUT: An instance of a @ref{7d,,FindStatStatistic}, represented by @itemize - @item the FindStat identifier together with its name, or @item a list of triples, each consisting of @quotation @itemize - @item the statistic @item a list of strings naming certain maps @item a number which says how many of the values submitted agree with the values in the database, when applying the maps in the given order to the object and then computing the statistic on the result. @end itemize @end quotation @end itemize EXAMPLES: A particular statistic can be retrieved by its St-identifier or number: @example sage: findstat('St000041') # optional -- internet,random St000041: The number of nestings of a perfect matching. sage: findstat(51) # optional -- internet,random St000051: The size of the left subtree. @end example The database can be searched by providing a list of pairs: @example sage: q = findstat([(pi, pi.length()) for pi in Permutations(4)]); q # optional -- internet,random 0: (St000018: The [[/Permutations/Inversions|number of inversions]] of a permutation., [], 24) 1: (St000004: The [[/Permutations/Descents-Major|major index]] of a permutation., [Mp00062: inversion-number to major-index bijection], 24) ... @end example or a dictionary: @example sage: p = findstat(@{pi: pi.length() for pi in Permutations(4)@}); p # optional -- internet,random 0: (St000018: The [[/Permutations/Inversions|number of inversions]] of a permutation., [], 24) 1: (St000004: The [[/Permutations/Descents-Major|major index]] of a permutation., [Mp00062: inversion-number to major-index bijection], 24) ... @end example Note however, that the results of these two queries are not necessarily the same, because we compare queries by the data sent, and the ordering of the data might be different: @example sage: p == q # optional -- internet False @end example Another possibility is to send a collection and a function. In this case, the function is applied to the first few objects of the collection: @example sage: findstat("Permutations", lambda pi: pi.length()) # optional -- internet,random 0: (St000018: The [[/Permutations/Inversions|number of inversions]] of a permutation., [], 200) ... @end example To search for a distribution, send a list of lists, or a single pair: @example sage: S4 = Permutations(4); findstat((S4, [pi.length() for pi in S4])) # optional -- internet,random 0: (St000004: The [[/Permutations/Descents-Major|major index]] of a permutation., [], 24) 1: (St000018: The [[/Permutations/Inversions|number of inversions]] of a permutation., [], 24) ... @end example Note that there is a limit, @code{FINDSTAT_MAX_DEPTH}, on the number of elements that may be submitted to FindStat, which is currently 200. Therefore, the interface tries to truncate queries appropriately, but this may be impossible, especially with distribution searches: @example sage: S6 = Permutations(6); S6.cardinality() # optional -- internet 720 sage: findstat((S6, [1 for a in S6])) # optional -- internet Traceback (most recent call last): ... ValueError: After discarding elements not in the range, too few (=0) values remained to send to FindStat. @end example @geindex browse() (sage.databases.findstat.FindStat method) @anchor{sage/databases/findstat sage databases findstat FindStat browse}@anchor{85} @deffn {Method} browse () Open the FindStat web page in a browser. EXAMPLES: @example sage: findstat.browse() # optional -- webbrowser @end example @end deffn @geindex login() (sage.databases.findstat.FindStat method) @anchor{sage/databases/findstat sage databases findstat FindStat login}@anchor{86} @deffn {Method} login () Open the FindStat login page in a browser. EXAMPLES: @example sage: findstat.login() # optional -- webbrowser @end example @end deffn @geindex set_user() (sage.databases.findstat.FindStat method) @anchor{sage/databases/findstat sage databases findstat FindStat set_user}@anchor{87} @deffn {Method} set_user (name=None, email=None) Set the user for this session. INPUT: @itemize - @item @code{name} -- the name of the user. @item @code{email} -- an email address of the user. @end itemize This information is used when submitting a statistic with @ref{80,,FindStatStatistic.submit()}. EXAMPLES: @example sage: findstat.set_user(name="Anonymous", email="invalid@@org") @end example @cartouche @quotation Note It is usually more convenient to login into the FindStat web page using the @ref{86,,login()} method. @end quotation @end cartouche @end deffn @end deffn @geindex FindStatCollection (class in sage.databases.findstat) @anchor{sage/databases/findstat sage databases findstat FindStatCollection}@anchor{88} @deffn {Class} sage.databases.findstat.FindStatCollection (parent, id, c, sageconstructor_overridden) Bases: sage.structure.element.Element@footnote{../../../../../../html/en/reference/structure/sage/structure/element.html#sage.structure.element.Element} A FindStat collection. @ref{88,,FindStatCollection} is a class representing a combinatorial collection available in the FindStat database. Its main use is to allow easy specification of the combinatorial collection when using @ref{7a,,findstat}. It also provides methods to quickly access its FindStat web page (@ref{89,,browse()}), check whether a particular element is actually in the range considered by FindStat (@ref{8a,,in_range()}), etc. INPUT: One of the following: @itemize - @item a string eg. 'Dyck paths' or 'DyckPaths', case-insensitive, or @item an integer designating the FindStat id of the collection, or @item a sage object belonging to a collection, or @item an iterable producing a sage object belonging to a collection. @end itemize EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection("Dyck paths") # optional -- internet Cc0005: Dyck paths sage: FindStatCollection(5) # optional -- internet Cc0005: Dyck paths sage: FindStatCollection(DyckWord([1,0,1,0])) # optional -- internet Cc0005: Dyck paths sage: FindStatCollection(DyckWords(2)) # optional -- internet Cc0005: Dyck paths @end example SEEALSO: @ref{8b,,FindStatCollections} @geindex browse() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection browse}@anchor{89} @deffn {Method} browse () Open the FindStat web page of the collection in a browser. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection("Permutations").browse() # optional -- webbrowser @end example @end deffn @geindex first_terms() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection first_terms}@anchor{8c} @deffn {Method} first_terms (statistic, max_values=1200) Compute the first few terms of the given statistic. INPUT: @itemize - @item @code{statistic} -- a callable. @item @code{max_values} -- the number of terms to compute at most. @end itemize OUTPUT: A list of pairs of the form (object, value). EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet sage: c.first_terms(lambda x: 1, max_values=10) # optional -- internet,random [([[0]], 1), ([[1]], 1), ([[2]], 1), ([[3]], 1), ([[0, 0], [0]], 1), ([[1, 0], [0]], 1), ([[1, 0], [1]], 1), ([[1, 1], [1]], 1), ([[2, 0], [0]], 1), ([[2, 0], [1]], 1)] @end example @end deffn @geindex from_string() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection from_string}@anchor{8d} @deffn {Method} from_string () Return a function that returns the object given the FindStat normal representation. OUTPUT: The function that produces the sage object given its FindStat normal representation as a string. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("Posets") # optional -- internet sage: p = c.from_string()('([(0, 2), (2, 1)], 3)') # optional -- internet sage: p.cover_relations() # optional -- internet [[0, 2], [2, 1]] sage: c = FindStatCollection("Binary Words") # optional -- internet sage: w = c.from_string()('010101') # optional -- internet sage: w in c._sageconstructor(6) # optional -- internet True @end example @end deffn @geindex id() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection id}@anchor{8e} @deffn {Method} id () Return the FindStat identifier of the collection. OUTPUT: The FindStat identifier of the collection as an integer. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet sage: c.id() # optional -- internet 18 @end example @end deffn @geindex id_str() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection id_str}@anchor{8f} @deffn {Method} id_str () Return the FindStat identifier of the collection. OUTPUT: The FindStat identifier of the collection as a string. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet sage: c.id_str() # optional -- internet 'Cc0018' @end example @end deffn @geindex in_range() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection in_range}@anchor{8a} @deffn {Method} in_range (element) Check whether an element of the collection is in FindStat's precomputed range. INPUT: @itemize - @item @code{element} -- a sage object that belongs to the collection. @end itemize OUTPUT: @code{True}, if @code{element} is used by the FindStat search engine, and @code{False} if it is ignored. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet sage: c.in_range(GelfandTsetlinPattern([[2, 1], [1]])) # optional -- internet True sage: c.in_range(GelfandTsetlinPattern([[3, 1], [1]])) # optional -- internet True sage: c.in_range(GelfandTsetlinPattern([[4, 1], [1]])) # optional -- internet,random False @end example TESTS: @example sage: from sage.databases.findstat import FindStatCollections sage: l = FindStatCollections() # optional -- internet sage: long = [9, 12, 14, 20] sage: for c in l: # optional -- internet, random ....: if c.id() not in long and c.is_supported(): ....: f = c.first_terms(lambda x: 1, max_values=10000) ....: print("@{@} @{@} @{@}".format(c, len(f), all(c.in_range(e) for e, _ in f))) ....: Cc0001: Permutations 10000 True Cc0002: Integer partitions 270 True Cc0005: Dyck paths 2054 True Cc0006: Integer compositions 510 True Cc0007: Standard tableaux 3734 True Cc0010: Binary trees 2054 True Cc0013: Cores 100 True Cc0017: Alternating sign matrices 7916 True Cc0018: Gelfand-Tsetlin patterns 934 True Cc0019: Semistandard tableaux 10000 True Cc0021: Ordered trees 2055 True Cc0022: Finite Cartan types 31 True Cc0023: Parking functions 10000 True @end example @end deffn @geindex is_supported() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection is_supported}@anchor{90} @deffn {Method} is_supported () Check whether the collection is fully supported by the interface. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection(1).is_supported() # optional -- internet True sage: FindStatCollection(24).is_supported() # optional -- internet, random False @end example @end deffn @geindex name() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection name}@anchor{91} @deffn {Method} name (style='singular') Return the name of the FindStat collection. INPUT: @itemize - @item a string -- (default:"singular") can be "singular", or "plural". @end itemize OUTPUT: The name of the FindStat collection, in singular or in plural. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection("Binary trees").name() # optional -- internet u'Binary tree' sage: FindStatCollection("Binary trees").name(style="plural") # optional -- internet u'Binary trees' @end example @end deffn @geindex to_string() (sage.databases.findstat.FindStatCollection method) @anchor{sage/databases/findstat sage databases findstat FindStatCollection to_string}@anchor{92} @deffn {Method} to_string () Return a function that returns the FindStat normal representation given an object. OUTPUT: The function that produces the string representation as needed by the FindStat search webpage. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollection sage: p = Poset((range(3), [[0, 1], [1, 2]])) # optional -- internet sage: c = FindStatCollection("Posets") # optional -- internet sage: c.to_string()(p) # optional -- internet '([(0, 2), (2, 1)], 3)' @end example @end deffn @end deffn @geindex FindStatCollections (class in sage.databases.findstat) @anchor{sage/databases/findstat sage databases findstat FindStatCollections}@anchor{8b} @deffn {Class} sage.databases.findstat.FindStatCollections Bases: sage.structure.parent.Parent@footnote{../../../../../../html/en/reference/structure/sage/structure/parent.html#sage.structure.parent.Parent}, sage.structure.unique_representation.UniqueRepresentation@footnote{../../../../../../html/en/reference/structure/sage/structure/unique_representation.html#sage.structure.unique_representation.UniqueRepresentation} The class of FindStat collections. The elements of this class are combinatorial collections in FindStat as of August 2015. If a new collection was added to the web service since then, the dictionary @code{_findstat_collections} in this class has to be updated accordingly. EXAMPLES: @example sage: from sage.databases.findstat import FindStatCollections sage: sorted(c for c in FindStatCollections()) # optional -- internet, random [Cc0001: Permutations, Cc0002: Integer partitions, Cc0005: Dyck paths, Cc0006: Integer compositions, Cc0007: Standard tableaux, Cc0009: Set partitions, Cc0010: Binary trees, Cc0012: Perfect matchings, Cc0013: Cores, Cc0014: Posets, Cc0017: Alternating sign matrices, Cc0018: Gelfand-Tsetlin patterns, Cc0019: Semistandard tableaux, Cc0020: Graphs, Cc0021: Ordered trees, Cc0022: Finite Cartan types, Cc0023: Parking functions] @end example @geindex Element (sage.databases.findstat.FindStatCollections attribute) @anchor{sage/databases/findstat sage databases findstat FindStatCollections Element}@anchor{93} @deffn {Attribute} Element alias of @ref{88,,FindStatCollection} @end deffn @end deffn @geindex FindStatMap (class in sage.databases.findstat) @anchor{sage/databases/findstat sage databases findstat FindStatMap}@anchor{7e} @deffn {Class} sage.databases.findstat.FindStatMap (parent, entry) Bases: sage.structure.element.Element@footnote{../../../../../../html/en/reference/structure/sage/structure/element.html#sage.structure.element.Element} A FindStat map. @ref{7e,,FindStatMap} is a class representing a combinatorial map available in the FindStat database. The result of a @ref{7a,,findstat} query contains a (possibly empty) list of such maps. This class provides methods to inspect various properties of these maps, in particular @ref{94,,code()}. INPUT: @itemize - @item a string containing the FindStat name of the map, or an integer representing its FindStat id. @end itemize EXAMPLES: @example sage: from sage.databases.findstat import FindStatMap sage: FindStatMap(71) # optional -- internet Mp00071: descent composition sage: FindStatMap("descent composition") # optional -- internet Mp00071: descent composition @end example SEEALSO: @ref{95,,FindStatMaps} @geindex code() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap code}@anchor{94} @deffn {Method} code () Return the code associated with the map. OUTPUT: A string. EXAMPLES: @example sage: from sage.databases.findstat import FindStatMap # optional -- internet sage: print(FindStatMap(71).code()) # optional -- internet def descents_composition(elt): if len(elt) == 0: return Composition([]) d = [-1] + elt.descents() + [len(elt)-1] return Composition([ d[i+1]-d[i] for i in range(len(d)-1)]) @end example @end deffn @geindex code_name() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap code_name}@anchor{96} @deffn {Method} code_name () Return the name of the function defined by @ref{94,,code()}. OUTPUT: A string. EXAMPLES: @example sage: from sage.databases.findstat import FindStatMap # optional -- internet sage: print(FindStatMap(71).code_name()) # optional -- internet descents_composition @end example @end deffn @geindex codomain() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap codomain}@anchor{97} @deffn {Method} codomain () Return the FindStat collection which is the codomain of the map. OUTPUT: The codomain of the map as a @ref{88,,FindStatCollection}. EXAMPLES: @example sage: from sage.databases.findstat import FindStatMap # optional -- internet sage: FindStatMap(71).codomain() # optional -- internet Cc0006: Integer compositions @end example @end deffn @geindex description() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap description}@anchor{98} @deffn {Method} description () Return the FindStat description of the map. OUTPUT: The description as a string. EXAMPLES: @example sage: m = findstat("Permutations", lambda pi: pi.length())[1][1][0] # optional -- internet sage: print(m.description()) # optional -- internet,random Let $\sigma \in \mathcal@{S@}_n$ be a permutation. Maps $\sigma$ to the permutation $\tau$ such that the major code of $\tau$ is given by the Lehmer code of $\sigma$. In particular, the number of inversions of $\sigma$ equals the major index of $\tau$. EXAMPLES: $[3,4,1,2] \mapsto [3,1,4,2]$ @end example @end deffn @geindex domain() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap domain}@anchor{99} @deffn {Method} domain () Return the FindStat collection which is the domain of the map. OUTPUT: The domain of the map as a @ref{88,,FindStatCollection}. EXAMPLES: @example sage: from sage.databases.findstat import FindStatMap # optional -- internet sage: FindStatMap(71).domain() # optional -- internet Cc0001: Permutations @end example @end deffn @geindex id() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap id}@anchor{9a} @deffn {Method} id () Return the FindStat identifier of the map. OUTPUT: The FindStat identifier of the map as an integer. EXAMPLES: @example sage: m = findstat("Permutations", lambda pi: pi.length())[1][1][0] # optional -- internet sage: m.id() # optional -- internet 62 @end example @end deffn @geindex id_str() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap id_str}@anchor{9b} @deffn {Method} id_str () Return the FindStat identifier of the map. OUTPUT: The FindStat identifier of the map as a string. EXAMPLES: @example sage: m = findstat("Permutations", lambda pi: pi.length())[1][1][0] # optional -- internet sage: m.id_str() # optional -- internet 'Mp00062' @end example @end deffn @geindex name() (sage.databases.findstat.FindStatMap method) @anchor{sage/databases/findstat sage databases findstat FindStatMap name}@anchor{9c} @deffn {Method} name () Return the FindStat name of the map. OUTPUT: The name of the map as a string, as used by FindStat. EXAMPLES: @example sage: m = findstat("Permutations", lambda pi: pi.length())[1][1][0] # optional -- internet sage: m.name() # optional -- internet u'inversion-number to major-index bijection' @end example @end deffn @end deffn @geindex FindStatMaps (class in sage.databases.findstat) @anchor{sage/databases/findstat sage databases findstat FindStatMaps}@anchor{95} @deffn {Class} sage.databases.findstat.FindStatMaps Bases: sage.structure.parent.Parent@footnote{../../../../../../html/en/reference/structure/sage/structure/parent.html#sage.structure.parent.Parent}, sage.structure.unique_representation.UniqueRepresentation@footnote{../../../../../../html/en/reference/structure/sage/structure/unique_representation.html#sage.structure.unique_representation.UniqueRepresentation} The class of FindStat maps. The elements of this class are combinatorial maps currently in FindStat. EXAMPLES: We can print a nice list of maps currently in FindStat, sorted by domain and codomain: @example sage: from sage.databases.findstat import FindStatMap, FindStatMaps sage: for m in sorted(FindStatMaps(), key=lambda m: (m.domain(), m.codomain)): # optional -- internet,random ....: print(m.domain().name().ljust(30)+" "+m.codomain().name().ljust(30)+" "+m.name()) Permutation Standard tableau Robinson-Schensted insertion tableau Permutation Integer partition Robinson-Schensted tableau shape Permutation Binary tree to increasing tree ... @end example @geindex Element (sage.databases.findstat.FindStatMaps attribute) @anchor{sage/databases/findstat sage databases findstat FindStatMaps Element}@anchor{9d} @deffn {Attribute} Element alias of @ref{7e,,FindStatMap} @end deffn @end deffn @geindex FindStatStatistic (class in sage.databases.findstat) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic}@anchor{7d} @deffn {Class} sage.databases.findstat.FindStatStatistic (id, first_terms=None, data=None, function=None, code='@w{'}, collection=None, depth=None) Bases: sage.structure.sage_object.SageObject@footnote{../../../../../../html/en/reference/structure/sage/structure/sage_object.html#sage.structure.sage_object.SageObject} The class of FindStat statistics. Do not instantiate this class directly. Instead, use @ref{7a,,findstat}. @geindex browse() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic browse}@anchor{9e} @deffn {Method} browse () Open the FindStat web page of the statistic in a browser. EXAMPLES: @example sage: findstat(41).browse() # optional -- webbrowser @end example @end deffn @geindex code() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic code}@anchor{9f} @deffn {Method} code () Return the code associated with the statistic. OUTPUT: A string. Contributors are encouraged to submit sage code in the form: @example def statistic(x): ... @end example but the string may also contain code for other computer algebra systems. EXAMPLES: @example sage: print(findstat(1).code()) # optional -- internet,random def statistic(x): return len(x.reduced_words()) sage: print(findstat(118).code()) # optional -- internet,random (* in Mathematica *) tree = @{@{@{@{@}, @{@}@}, @{@{@}, @{@}@}@}, @{@{@{@}, @{@}@}, @{@{@}, @{@}@}@}@}; Count[tree, @{@{___@}, @{@{___@}, @{@{___@}, @{___@}@}@}@}, @{0, Infinity@}] @end example @end deffn @geindex collection() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic collection}@anchor{a0} @deffn {Method} collection () Return the FindStat collection of the statistic. OUTPUT: The FindStat collection of the statistic as an instance of @ref{88,,FindStatCollection}. EXAMPLES: @example sage: findstat(1).collection() # optional -- internet Cc0001: Permutations @end example @end deffn @geindex data() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic data}@anchor{a1} @deffn {Method} data () Return the data used for querying the FindStat database. OUTPUT: The data provided by the user to query the FindStat database. When the database was searched using an identifier, @code{data} is @code{None}. EXAMPLES: @example sage: S4 = Permutations(4); findstat((S4, [pi.length() for pi in S4])).data() # optional -- internet [(Standard permutations of 4, ['[1, 2, 3, 4]', '[1, 2, 4, 3]', '[1, 3, 2, 4]', '[1, 3, 4, 2]', '[1, 4, 2, 3]', '[1, 4, 3, 2]', '[2, 1, 3, 4]', '[2, 1, 4, 3]', '[2, 3, 1, 4]', '[2, 3, 4, 1]', '[2, 4, 1, 3]', '[2, 4, 3, 1]', '[3, 1, 2, 4]', '[3, 1, 4, 2]', '[3, 2, 1, 4]', '[3, 2, 4, 1]', '[3, 4, 1, 2]', '[3, 4, 2, 1]', '[4, 1, 2, 3]', '[4, 1, 3, 2]', '[4, 2, 1, 3]', '[4, 2, 3, 1]', '[4, 3, 1, 2]', '[4, 3, 2, 1]'], [0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6])] @end example @end deffn @geindex description() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic description}@anchor{a2} @deffn {Method} description () Return the description of the statistic. OUTPUT: A string, whose first line is used as the name of the statistic. EXAMPLES: @example sage: print(findstat(1).description()) # optional -- internet,random The number of ways to write a permutation as a minimal length product of simple transpositions. That is, the number of reduced words for the permutation. E.g., there are two reduced words for $[3,2,1] = (1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$. @end example @end deffn @geindex edit() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic edit}@anchor{a3} @deffn {Method} edit (max_values=1200) Open the FindStat web page for editing the statistic in a browser. INPUT: @itemize - @item @code{max_values} -- integer (default: @code{FINDSTAT_MAX_SUBMISSION_VALUES}); if @ref{a4,,function()} is defined and the statistic is a new statistic, use @ref{8c,,FindStatCollection.first_terms()} to produce at most @code{max_values} terms. @end itemize OUTPUT: @itemize - @item Raise an error if the query has a match with no intermediate combinatorial maps. @end itemize EXAMPLES: @example sage: s = findstat(DyckWords(4), lambda x: randint(1,1000)); s # optional -- internet a new statistic on Cc0005: Dyck paths @end example The following uses @code{lambda x: randint(1,1000)} to produce 14 terms, because @code{min(DyckWords(4).cardinality(), FINDSTAT_MAX_SUBMISSION_VALUES)} is 14: @example sage: s.submit() # optional -- webbrowser @end example @end deffn @geindex first_terms() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic first_terms}@anchor{a5} @deffn {Method} first_terms () Return the first terms of the statistic. OUTPUT: A list of pairs of the form @code{(object, value)} where @code{object} is a sage object representing an element of the appropriate collection and @code{value} is an integer. If the statistic is in the FindStat database, the list contains exactly the pairs in the database. EXAMPLES: @example sage: findstat(1).first_terms() # optional -- internet,random [([1], 1), ([1, 2], 1), ([2, 1], 1), ([1, 2, 3], 1), ([1, 3, 2], 1), ([2, 1, 3], 1), ... @end example TESTS: @example sage: r = findstat(@{d: randint(1,1000) for d in DyckWords(4)@}); r # optional -- internet a new statistic on Cc0005: Dyck paths sage: isinstance(r.first_terms(), list) # optional -- internet True sage: all(isinstance(e, tuple) and len(e)==2 and isinstance(e[1], (ZZ, Integer, int)) for e in r.first_terms()) # optional -- internet True @end example @end deffn @geindex first_terms_str() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic first_terms_str}@anchor{a6} @deffn {Method} first_terms_str () Return the first terms of the statistic in the format needed for a FindStat query. OUTPUT: A string, where each line is of the form @code{object => value}, where @code{object} is the string representation of an element of the appropriate collection as used by FindStat and value is an integer. EXAMPLES: @example sage: findstat(1).first_terms_str()[:10] # optional -- internet,random '[1] => 1\r\n' @end example @end deffn @geindex function() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic function}@anchor{a4} @deffn {Method} function () Return the function used to compute the values of the statistic. OUTPUT: The function used to compute the values of the statistic, or @code{None}. EXAMPLES: @example sage: findstat("Permutations", lambda pi: pi.length()).function() # optional -- internet ... at ...> @end example @end deffn @geindex generating_functions() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic generating_functions}@anchor{a7} @deffn {Method} generating_functions (style='polynomial') Return the generating functions of @code{self} in a dictionary. The keys of this dictionary are the levels for which the generating function of @code{self} can be computed from the data of this statistic, and each value represents a generating function for one level, as a polynomial, as a dictionary, or as a list of coefficients. INPUT: @itemize - @item a string -- (default:"polynomial") can be "polynomial", "dictionary", or "list". @end itemize OUTPUT: @itemize - @item if @code{style} is @code{"polynomial"}, the generating function is returned as a polynomial. @item if @code{style} is @code{"dictionary"}, the generating function is returned as a dictionary representing the monomials of the generating function. @item if @code{style} is @code{"list"}, the generating function is returned as a list of coefficients of the generating function. @end itemize EXAMPLES: @example sage: st = findstat(18) # optional -- internet sage: st.generating_functions() # optional -- internet,random @{2: q + 1, 3: q^3 + 2*q^2 + 2*q + 1, 4: q^6 + 3*q^5 + 5*q^4 + 6*q^3 + 5*q^2 + 3*q + 1, 5: q^10 + 4*q^9 + 9*q^8 + 15*q^7 + 20*q^6 + 22*q^5 + 20*q^4 + 15*q^3 + 9*q^2 + 4*q + 1, 6: q^15 + 5*q^14 + 14*q^13 + 29*q^12 + 49*q^11 + 71*q^10 + 90*q^9 + 101*q^8 + 101*q^7 + 90*q^6 + 71*q^5 + 49*q^4 + 29*q^3 + 14*q^2 + 5*q + 1@} sage: st.generating_functions(style="dictionary") # optional -- internet,random @{2: @{0: 1, 1: 1@}, 3: @{0: 1, 1: 2, 2: 2, 3: 1@}, 4: @{0: 1, 1: 3, 2: 5, 3: 6, 4: 5, 5: 3, 6: 1@}, 5: @{0: 1, 1: 4, 2: 9, 3: 15, 4: 20, 5: 22, 6: 20, 7: 15, 8: 9, 9: 4, 10: 1@}, 6: @{0: 1, 1: 5, 2: 14, 3: 29, 4: 49, 5: 71, 6: 90, 7: 101, 8: 101, 9: 90, 10: 71, 11: 49, 12: 29, 13: 14, 14: 5, 15: 1@}@} sage: st.generating_functions(style="list") # optional -- internet,random @{2: [1, 1], 3: [1, 2, 2, 1], 4: [1, 3, 5, 6, 5, 3, 1], 5: [1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1], 6: [1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1]@} @end example @end deffn @geindex id() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic id}@anchor{a8} @deffn {Method} id () Return the FindStat identifier of the statistic. OUTPUT: The FindStat identifier of the statistic (or 0), as an integer. EXAMPLES: @example sage: findstat(1).id() # optional -- internet 1 @end example @end deffn @geindex id_str() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic id_str}@anchor{a9} @deffn {Method} id_str () Return the FindStat identifier of the statistic. OUTPUT: The FindStat identifier of the statistic (or 'St000000'), as a string. EXAMPLES: @example sage: findstat(1).id_str() # optional -- internet 'St000001' @end example @end deffn @geindex modified() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic modified}@anchor{aa} @deffn {Method} modified () Return whether the statistic was modified. OUTPUT: True, if the statistic was modified using @ref{81,,set_description()}, @ref{82,,set_code()}, @ref{83,,set_references()}, etc. False otherwise. EXAMPLES: @example sage: findstat(41).set_description("") # optional -- internet sage: findstat(41).modified() # optional -- internet True @end example @end deffn @geindex name() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic name}@anchor{ab} @deffn {Method} name () Return the name of the statistic. OUTPUT: A string, which is just the first line of the description of the statistic. EXAMPLES: @example sage: findstat(1).name() # optional -- internet,random u'The number of ways to write a permutation as a minimal length product of simple transpositions.' @end example @end deffn @geindex oeis_search() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic oeis_search}@anchor{ac} @deffn {Method} oeis_search (search_size=32, verbose=True) Search the OEIS for the generating function of the statistic. INPUT: @itemize - @item @code{search_size} (default:32) the number of integers in the sequence. If too big, the OEIS result is corrupted. @item @code{verbose} (default:True) if true, some information about the search are printed. @end itemize OUTPUT: @itemize - @item a tuple of OEIS sequences, see @ref{4c,,sage.databases.oeis.OEIS.find_by_description()} for more information. @end itemize EXAMPLES: @example sage: st = findstat(18) # optional -- internet sage: st.oeis_search() # optional -- internet,random Searching the OEIS for "1,1 1,2,2,1 1,3,5,6,5,3,1 1,4,9,15,20,22,20,15,9,4,1 1,5,14,29,49,71,90,101" 0: A008302: Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_@{i=0..n-1@} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). sage: st.oeis_search(search_size=13) # optional -- internet,random Searching the OEIS for "1,1 1,2,2,1 1,3,5,6,5,3,1" 0: A008302: Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_@{i=0..n-1@} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). 1: A115570: Array read by rows: row n (n>= 1) gives the Betti numbers for the n-th element of the Weyl group of type A3 (in Goresky's standard ordering). 2: A187447: Array for all multiset choices (multiset repetition class representatives in Abramowitz-Stegun order). @end example @end deffn @geindex references() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic references}@anchor{ad} @deffn {Method} references () Return the references associated with the statistic. OUTPUT: An instance of @ref{49,,sage.databases.oeis.FancyTuple}, each item corresponds to a reference. @cartouche @quotation Todo Since the references in the database are sometimes not formatted properly, this method is unreliable. The string representation can be obtained via @code{_references}. @end quotation @end cartouche EXAMPLES: @example sage: findstat(1).references() # optional -- internet,random 0: P. Edelman and C. Greene, Balanced tableaux, Adv. in Math., 63 (1987), pp. 42-99. 1: [[OEIS:A005118]] 2: [[oeis:A246865]] @end example @end deffn @geindex set_code() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic set_code}@anchor{82} @deffn {Method} set_code (value) Set the code associated with the statistic. INPUT: @itemize - @item a string -- contributors are encouraged to submit sage code in the form: @example def statistic(x): ... @end example @end itemize OUTPUT: @itemize - @item Raise an error if the query has a match with no intermediate combinatorial maps. @end itemize This information is used when submitting the statistic with @ref{80,,submit()}. EXAMPLES: @example sage: s = findstat([(d, randint(1,1000)) for d in DyckWords(4)]) # optional -- internet sage: s.set_code("def statistic(x):\r\n return randint(1,1000)") # optional -- internet sage: print(s.code()) # optional -- internet def statistic(x): return randint(1,1000) @end example @end deffn @geindex set_description() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic set_description}@anchor{81} @deffn {Method} set_description (value) Set the description of the statistic. INPUT: @itemize - @item a string -- the name of the statistic followed by its description on a separate line. @end itemize OUTPUT: @itemize - @item Raise an error, if the query has a match with no intermediate combinatorial maps. @end itemize This information is used when submitting the statistic with @ref{80,,submit()}. EXAMPLES: @example sage: s = findstat([(d, randint(1,1000)) for d in DyckWords(4)]); s # optional -- internet a new statistic on Cc0005: Dyck paths sage: s.set_description("Random values on Dyck paths.\r\nNot for submission.") # optional -- internet sage: s # optional -- internet a new statistic on Cc0005: Dyck paths sage: s.name() # optional -- internet 'Random values on Dyck paths.' sage: print(s.description()) # optional -- internet Random values on Dyck paths. Not for submission. @end example @end deffn @geindex set_references() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic set_references}@anchor{83} @deffn {Method} set_references (value) Set the references associated with the statistic. INPUT: @itemize - @item a string -- the individual references should be separated by FINDSTAT_SEPARATOR_REFERENCES, which is "\r\n". @end itemize OUTPUT: @itemize - @item Raise an error, if the query has a match with no intermediate combinatorial maps. @end itemize This information is used when submitting the statistic with @ref{80,,submit()}. EXAMPLES: @example sage: s = findstat([(d, randint(1,1000)) for d in DyckWords(4)]); s # optional -- internet a new statistic on Cc0005: Dyck paths sage: s.set_references("[1] The wonders of random Dyck paths, Anonymous Coward, [[arXiv:1102.4226]].\r\n[2] [[oeis:A000001]]") # optional -- internet sage: s.references() # optional -- internet 0: [1] The wonders of random Dyck paths, Anonymous Coward, [[arXiv:1102.4226]]. 1: [2] [[oeis:A000001]] @end example @end deffn @geindex submit() (sage.databases.findstat.FindStatStatistic method) @anchor{sage/databases/findstat sage databases findstat FindStatStatistic submit}@anchor{80} @deffn {Method} submit (max_values=1200) Open the FindStat web page for editing the statistic in a browser. INPUT: @itemize - @item @code{max_values} -- integer (default: @code{FINDSTAT_MAX_SUBMISSION_VALUES}); if @ref{a4,,function()} is defined and the statistic is a new statistic, use @ref{8c,,FindStatCollection.first_terms()} to produce at most @code{max_values} terms. @end itemize OUTPUT: @itemize - @item Raise an error if the query has a match with no intermediate combinatorial maps. @end itemize EXAMPLES: @example sage: s = findstat(DyckWords(4), lambda x: randint(1,1000)); s # optional -- internet a new statistic on Cc0005: Dyck paths @end example The following uses @code{lambda x: randint(1,1000)} to produce 14 terms, because @code{min(DyckWords(4).cardinality(), FINDSTAT_MAX_SUBMISSION_VALUES)} is 14: @example sage: s.submit() # optional -- webbrowser @end example @end deffn @end deffn @c nodoctest @node Frank Luebeck's tables of Conway polynomials over finite fields,Tables of zeros of the Riemann-Zeta function,FindStat - the Combinatorial Statistic Finder,Top @anchor{sage/databases/conway frank-luebeck-s-tables-of-conway-polynomials-over-finite-fields}@anchor{ae}@anchor{sage/databases/conway sage-databases-conway}@anchor{af}@anchor{sage/databases/conway doc}@anchor{b0} @chapter Frank Luebeck's tables of Conway polynomials over finite fields @c This file has been autogenerated. @anchor{sage/databases/conway module-sage databases conway}@anchor{0} @geindex sage.databases.conway (module) @geindex ConwayPolynomials (class in sage.databases.conway) @anchor{sage/databases/conway sage databases conway ConwayPolynomials}@anchor{b1} @deffn {Class} sage.databases.conway.ConwayPolynomials Bases: @code{_abcoll.Mapping} Initialize the database. TESTS: @example sage: c = ConwayPolynomials() sage: c Frank Luebeck's database of Conway polynomials @end example @geindex degrees() (sage.databases.conway.ConwayPolynomials method) @anchor{sage/databases/conway sage databases conway ConwayPolynomials degrees}@anchor{b2} @deffn {Method} degrees (p) Return the list of integers @code{n} for which the database of Conway polynomials contains the polynomial of degree @code{n} over @code{GF(p)}. EXAMPLES: @example sage: c = ConwayPolynomials() sage: c.degrees(60821) [1, 2, 3, 4] sage: c.degrees(next_prime(10^7)) [] @end example @end deffn @geindex has_polynomial() (sage.databases.conway.ConwayPolynomials method) @anchor{sage/databases/conway sage databases conway ConwayPolynomials has_polynomial}@anchor{b3} @deffn {Method} has_polynomial (p, n) Return True if the database of Conway polynomials contains the polynomial of degree @code{n} over @code{GF(p)}. INPUT: @itemize - @item @code{p} -- prime number @item @code{n} -- positive integer @end itemize EXAMPLES: @example sage: c = ConwayPolynomials() sage: c.has_polynomial(97, 12) True sage: c.has_polynomial(60821, 5) False @end example @end deffn @geindex polynomial() (sage.databases.conway.ConwayPolynomials method) @anchor{sage/databases/conway sage databases conway ConwayPolynomials polynomial}@anchor{b4} @deffn {Method} polynomial (p, n) Return the Conway polynomial of degree @code{n} over @code{GF(p)}, or raise a RuntimeError if this polynomial is not in the database. @cartouche @quotation Note See also the global function @code{conway_polynomial} for a more user-friendly way of accessing the polynomial. @end quotation @end cartouche INPUT: @itemize - @item @code{p} -- prime number @item @code{n} -- positive integer @end itemize OUTPUT: List of Python int's giving the coefficients of the corresponding Conway polynomial in ascending order of degree. EXAMPLES: @example sage: c = ConwayPolynomials() sage: c.polynomial(3, 21) (1, 2, 0, 2, 0, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1) sage: c.polynomial(97, 128) Traceback (most recent call last): ... RuntimeError: Conway polynomial over F_97 of degree 128 not in database. @end example @end deffn @geindex primes() (sage.databases.conway.ConwayPolynomials method) @anchor{sage/databases/conway sage databases conway ConwayPolynomials primes}@anchor{b5} @deffn {Method} primes () Return the list of prime numbers @code{p} for which the database of Conway polynomials contains polynomials over @code{GF(p)}. EXAMPLES: @example sage: c = ConwayPolynomials() sage: P = c.primes() sage: 2 in P True sage: next_prime(10^7) in P False @end example @end deffn @end deffn @geindex DictInMapping (class in sage.databases.conway) @anchor{sage/databases/conway sage databases conway DictInMapping}@anchor{b6} @deffn {Class} sage.databases.conway.DictInMapping (dict) Bases: @code{_abcoll.Mapping} Places dict into a non-mutable mapping. TESTS: @example sage: from sage.databases.conway import DictInMapping sage: d = @{@} sage: m = DictInMapping(d); m @{@} sage: d[0] = 1; m @{0: 1@} sage: m[2] = 3 Traceback (most recent call last): ... TypeError: 'DictInMapping' object does not support item assignment @end example @end deffn @c nodoctest @node Tables of zeros of the Riemann-Zeta function,Ideals from the Symbolic Data project,Frank Luebeck's tables of Conway polynomials over finite fields,Top @anchor{sage/databases/odlyzko sage-databases-odlyzko}@anchor{b7}@anchor{sage/databases/odlyzko doc}@anchor{b8}@anchor{sage/databases/odlyzko tables-of-zeros-of-the-riemann-zeta-function}@anchor{b9} @chapter Tables of zeros of the Riemann-Zeta function @c This file has been autogenerated. @anchor{sage/databases/odlyzko module-sage databases odlyzko}@anchor{7} @geindex sage.databases.odlyzko (module) AUTHORS: @itemize - @item William Stein: initial version @item Jeroen Demeyer (2015-01-20): convert @code{database_odlyzko_zeta} to new-style package @end itemize @geindex zeta_zeros() (in module sage.databases.odlyzko) @anchor{sage/databases/odlyzko sage databases odlyzko zeta_zeros}@anchor{ba} @deffn {Function} sage.databases.odlyzko.zeta_zeros () List of the imaginary parts of the first 2,001,052 zeros of the Riemann zeta function, accurate to within 4e-9. In order to use @code{zeta_zeros()}, you will need to install the optional Odlyzko database package: @example sage -i database_odlyzko_zeta @end example You can see a list of all available optional packages with @code{sage --optional}. REFERENCES: @itemize - @item @indicateurl{http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html} @end itemize EXAMPLES: The following example prints the imaginary part of the 13th nontrivial zero of the Riemann zeta function: @example sage: zz = zeta_zeros() # optional - database_odlyzko_zeta sage: zz[12] # optional - database_odlyzko_zeta 59.347044003 sage: len(zz) # optional - database_odlyzko_zeta 2001052 @end example @end deffn @c nodoctest @node Ideals from the Symbolic Data project,Cunningham table,Tables of zeros of the Riemann-Zeta function,Top @anchor{sage/databases/symbolic_data doc}@anchor{bb}@anchor{sage/databases/symbolic_data ideals-from-the-symbolic-data-project}@anchor{bc}@anchor{sage/databases/symbolic_data sage-databases-symbolic-data}@anchor{bd} @chapter Ideals from the Symbolic Data project @c This file has been autogenerated. @anchor{sage/databases/symbolic_data module-sage databases symbolic_data}@anchor{b} @geindex sage.databases.symbolic_data (module) This file implements a thin wrapper for the optional symbolic data set of ideals as published on @indicateurl{http://www.symbolicdata.org} . From the project website: @quotation For different purposes algorithms and implementations are tested on certified and reliable data. The development of tools and data for such tests is usually 'orthogonal' to the main implementation efforts, it requires different skills and technologies and is not loved by programmers. On the other hand, in many cases tools and data could easily be reused - with slight modifications - across similar projects. The SymbolicData Project is set out to coordinate such efforts within the Computer Algebra Community. Commonly collected certified and reliable data can also be used to compare otherwise incomparable approaches, algorithms, and implementations. Benchmark suites and Challenges for symbolic computations are not as well established as in other areas of computer science. This is probably due to the fact that there are not yet well agreed aims of such a benchmarking. Nevertheless various (often high quality) special benchmarks are scattered through the literature. During the last years efforts toward collection of test data for symbolic computations were intensified. They focused mainly on the creation of general benchmarks for different areas of symbolic computation and the collection of such activities on different Web site. For further qualification of these efforts it would be of great benefit to create a commonly available digital archive of these special benchmark data scattered through the literature. This would provide the community with an electronic repository of certified data that could be addressed and extended during further development. @end quotation EXAMPLES: @example sage: sd = SymbolicData(); sd # optional - database_symbolic_data SymbolicData with 372 ideals sage: sd.ZeroDim__example_1 # optional - database_symbolic_data Ideal (x1^2 + x2^2 - 10, x1^2 + x1*x2 + 2*x2^2 - 16) of Multivariate Polynomial Ring in x1, x2 over Rational Field sage: sd.Katsura_3 # optional - database_symbolic_data Ideal (u0 + 2*u1 + 2*u2 + 2*u3 - 1, u1^2 + 2*u0*u2 + 2*u1*u3 - u2, 2*u0*u1 + 2*u1*u2 + 2*u2*u3 - u1, u0^2 + 2*u1^2 + 2*u2^2 + 2*u3^2 - u0) of Multivariate Polynomial Ring in u0, u1, u2, u3 over Rational Field sage: sd.get_ideal('Katsura_3',GF(127),'degrevlex') # optional - database_symbolic_data Ideal (u0 + 2*u1 + 2*u2 + 2*u3 - 1, u1^2 + 2*u0*u2 + 2*u1*u3 - u2, 2*u0*u1 + 2*u1*u2 + 2*u2*u3 - u1, u0^2 + 2*u1^2 + 2*u2^2 + 2*u3^2 - u0) of Multivariate Polynomial Ring in u0, u1, u2, u3 over Finite Field of size 127 @end example AUTHORS: @itemize - @item Martin Albrecht <@email{martinralbrecht@@googlemail.com}> @end itemize @geindex SymbolicData (class in sage.databases.symbolic_data) @anchor{sage/databases/symbolic_data sage databases symbolic_data SymbolicData}@anchor{be} @deffn {Class} sage.databases.symbolic_data.SymbolicData Database of ideals as distributed by the The SymbolicData Project (@indicateurl{http://symbolicdata.org}). This class needs the optional @code{database_symbolic_data} package to be installed. @geindex get_ideal() (sage.databases.symbolic_data.SymbolicData method) @anchor{sage/databases/symbolic_data sage databases symbolic_data SymbolicData get_ideal}@anchor{bf} @deffn {Method} get_ideal (name, base_ring=Rational Field, term_order='degrevlex') Returns the ideal given by 'name' over the base ring given by 'base_ring' in a polynomial ring with the term order given by 'term_order'. INPUT: @itemize - @item @code{name} - name as on the symbolic data website @item @code{base_ring} - base ring for the polynomial ring (default: @code{QQ}) @item @code{term_order} - term order for the polynomial ring (default: @code{degrevlex}) @end itemize OUTPUT: @quotation ideal as given by @code{name} in @code{PolynomialRing(base_ring,vars,term_order)} @end quotation EXAMPLES: @example sage: sd = SymbolicData() # optional - database_symbolic_data sage: sd.get_ideal('Katsura_3',GF(127),'degrevlex') # optional - database_symbolic_data Ideal (u0 + 2*u1 + 2*u2 + 2*u3 - 1, u1^2 + 2*u0*u2 + 2*u1*u3 - u2, 2*u0*u1 + 2*u1*u2 + 2*u2*u3 - u1, u0^2 + 2*u1^2 + 2*u2^2 + 2*u3^2 - u0) of Multivariate Polynomial Ring in u0, u1, u2, u3 over Finite Field of size 127 @end example @end deffn @geindex trait_names() (sage.databases.symbolic_data.SymbolicData method) @anchor{sage/databases/symbolic_data sage databases symbolic_data SymbolicData trait_names}@anchor{c0} @deffn {Method} trait_names () EXAMPLES: @example sage: sd = SymbolicData() # optional - database_symbolic_data sage: sorted(sd.trait_names())[:10] # optional - database_symbolic_data ['Bjoerk_8', 'Bronstein-86', 'Buchberger-87', 'Butcher', 'Caprasse', 'Cassou', 'Cohn_2', 'Curves__curve10_20', 'Curves__curve10_20', 'Curves__curve10_30'] @end example @end deffn @end deffn @c nodoctest @node Cunningham table,Database of Hilbert Polynomials,Ideals from the Symbolic Data project,Top @anchor{sage/databases/cunningham_tables sage-databases-cunningham-tables}@anchor{c1}@anchor{sage/databases/cunningham_tables doc}@anchor{c2}@anchor{sage/databases/cunningham_tables cunningham-table}@anchor{c3} @chapter Cunningham table @c This file has been autogenerated. @anchor{sage/databases/cunningham_tables module-sage databases cunningham_tables}@anchor{2} @geindex sage.databases.cunningham_tables (module) @geindex cunningham_prime_factors() (in module sage.databases.cunningham_tables) @anchor{sage/databases/cunningham_tables sage databases cunningham_tables cunningham_prime_factors}@anchor{c4} @deffn {Function} sage.databases.cunningham_tables.cunningham_prime_factors () List of all the prime numbers occuring in the so called Cunningham table. They occur in the factorization of numbers of type @math{b^n+1} or @math{b^n-1} with @math{b \in \@{2@comma{}3@comma{}5@comma{}6@comma{}7@comma{}10@comma{}11@comma{}12\@}}. Data from @indicateurl{http://cage.ugent.be/~jdemeyer/cunningham/} @end deffn @c nodoctest @node Database of Hilbert Polynomials,Database of Modular Polynomials,Cunningham table,Top @anchor{sage/databases/db_class_polynomials sage-databases-db-class-polynomials}@anchor{c5}@anchor{sage/databases/db_class_polynomials doc}@anchor{c6}@anchor{sage/databases/db_class_polynomials database-of-hilbert-polynomials}@anchor{c7} @chapter Database of Hilbert Polynomials @c This file has been autogenerated. @anchor{sage/databases/db_class_polynomials module-sage databases db_class_polynomials}@anchor{3} @geindex sage.databases.db_class_polynomials (module) @geindex AtkinClassPolynomialDatabase (class in sage.databases.db_class_polynomials) @anchor{sage/databases/db_class_polynomials sage databases db_class_polynomials AtkinClassPolynomialDatabase}@anchor{c8} @deffn {Class} sage.databases.db_class_polynomials.AtkinClassPolynomialDatabase Bases: @ref{c9,,sage.databases.db_class_polynomials.ClassPolynomialDatabase} The database of Atkin class polynomials. @end deffn @geindex ClassPolynomialDatabase (class in sage.databases.db_class_polynomials) @anchor{sage/databases/db_class_polynomials sage databases db_class_polynomials ClassPolynomialDatabase}@anchor{c9} @deffn {Class} sage.databases.db_class_polynomials.ClassPolynomialDatabase @end deffn @geindex DedekindEtaClassPolynomialDatabase (class in sage.databases.db_class_polynomials) @anchor{sage/databases/db_class_polynomials sage databases db_class_polynomials DedekindEtaClassPolynomialDatabase}@anchor{ca} @deffn {Class} sage.databases.db_class_polynomials.DedekindEtaClassPolynomialDatabase Bases: @ref{c9,,sage.databases.db_class_polynomials.ClassPolynomialDatabase} The database of Dedekind eta class polynomials. @end deffn @geindex HilbertClassPolynomialDatabase (class in sage.databases.db_class_polynomials) @anchor{sage/databases/db_class_polynomials sage databases db_class_polynomials HilbertClassPolynomialDatabase}@anchor{cb} @deffn {Class} sage.databases.db_class_polynomials.HilbertClassPolynomialDatabase Bases: @ref{c9,,sage.databases.db_class_polynomials.ClassPolynomialDatabase} The database of Hilbert class polynomials. EXAMPLES: @example sage: db = HilbertClassPolynomialDatabase() sage: db[-4] # optional - database_kohel x - 1728 sage: db[-7] # optional - database_kohel x + 3375 sage: f = db[-23]; f # optional - database_kohel x^3 + 3491750*x^2 - 5151296875*x + 12771880859375 sage: f.discriminant().factor() # optional - database_kohel -1 * 5^18 * 7^12 * 11^4 * 17^2 * 19^2 * 23 sage: db[-23] # optional - database_kohel x^3 + 3491750*x^2 - 5151296875*x + 12771880859375 @end example @end deffn @geindex WeberClassPolynomialDatabase (class in sage.databases.db_class_polynomials) @anchor{sage/databases/db_class_polynomials sage databases db_class_polynomials WeberClassPolynomialDatabase}@anchor{cc} @deffn {Class} sage.databases.db_class_polynomials.WeberClassPolynomialDatabase Bases: @ref{c9,,sage.databases.db_class_polynomials.ClassPolynomialDatabase} The database of Weber class polynomials. @end deffn @c nodoctest @node Database of Modular Polynomials,Indices and Tables,Database of Hilbert Polynomials,Top @anchor{sage/databases/db_modular_polynomials database-of-modular-polynomials}@anchor{cd}@anchor{sage/databases/db_modular_polynomials doc}@anchor{ce}@anchor{sage/databases/db_modular_polynomials sage-databases-db-modular-polynomials}@anchor{cf} @chapter Database of Modular Polynomials @c This file has been autogenerated. @anchor{sage/databases/db_modular_polynomials module-sage databases db_modular_polynomials}@anchor{4} @geindex sage.databases.db_modular_polynomials (module) @geindex AtkinModularCorrespondenceDatabase (class in sage.databases.db_modular_polynomials) @anchor{sage/databases/db_modular_polynomials sage databases db_modular_polynomials AtkinModularCorrespondenceDatabase}@anchor{d0} @deffn {Class} sage.databases.db_modular_polynomials.AtkinModularCorrespondenceDatabase Bases: @ref{d1,,sage.databases.db_modular_polynomials.ModularCorrespondenceDatabase} @end deffn @geindex AtkinModularPolynomialDatabase (class in sage.databases.db_modular_polynomials) @anchor{sage/databases/db_modular_polynomials sage databases db_modular_polynomials AtkinModularPolynomialDatabase}@anchor{d2} @deffn {Class} sage.databases.db_modular_polynomials.AtkinModularPolynomialDatabase Bases: @ref{d3,,sage.databases.db_modular_polynomials.ModularPolynomialDatabase} The database of modular polynomials Phi(x,j) for @math{X_0(p)}, where x is a function on invariant under the Atkin-Lehner invariant, with pole of minimal order at infinity. @end deffn @geindex ClassicalModularPolynomialDatabase (class in sage.databases.db_modular_polynomials) @anchor{sage/databases/db_modular_polynomials sage databases db_modular_polynomials ClassicalModularPolynomialDatabase}@anchor{d4} @deffn {Class} sage.databases.db_modular_polynomials.ClassicalModularPolynomialDatabase Bases: @ref{d3,,sage.databases.db_modular_polynomials.ModularPolynomialDatabase} The database of classical modular polynomials, i.e. the polynomials Phi_N(X,Y) relating the j-functions j(q) and j(q^N). @end deffn @geindex DedekindEtaModularCorrespondenceDatabase (class in sage.databases.db_modular_polynomials) @anchor{sage/databases/db_modular_polynomials sage databases db_modular_polynomials DedekindEtaModularCorrespondenceDatabase}@anchor{d5} @deffn {Class} sage.databases.db_modular_polynomials.DedekindEtaModularCorrespondenceDatabase Bases: @ref{d1,,sage.databases.db_modular_polynomials.ModularCorrespondenceDatabase} The database of modular correspondences in @math{X_0(p) imes X_0(p)}, where the model of the curves @math{X_0(p) = \Bold@{P@}^1} are specified by quotients of Dedekind's eta function. @end deffn @geindex DedekindEtaModularPolynomialDatabase (class in sage.databases.db_modular_polynomials) @anchor{sage/databases/db_modular_polynomials sage databases db_modular_polynomials DedekindEtaModularPolynomialDatabase}@anchor{d6} @deffn {Class} sage.databases.db_modular_polynomials.DedekindEtaModularPolynomialDatabase Bases: @ref{d3,,sage.databases.db_modular_polynomials.ModularPolynomialDatabase} The database of modular polynomials Phi_N(X,Y) relating a quotient of Dedekind eta functions, well-defined on X_0(N), relating x(q) and the j-function j(q). @end deffn @geindex ModularCorrespondenceDatabase (class in sage.databases.db_modular_polynomials) @anchor{sage/databases/db_modular_polynomials sage databases db_modular_polynomials ModularCorrespondenceDatabase}@anchor{d1} @deffn {Class} sage.databases.db_modular_polynomials.ModularCorrespondenceDatabase Bases: @ref{d3,,sage.databases.db_modular_polynomials.ModularPolynomialDatabase} @end deffn @geindex ModularPolynomialDatabase (class in sage.databases.db_modular_polynomials) @anchor{sage/databases/db_modular_polynomials sage databases db_modular_polynomials ModularPolynomialDatabase}@anchor{d3} @deffn {Class} sage.databases.db_modular_polynomials.ModularPolynomialDatabase @end deffn @node Indices and Tables,Python Module Index,Database of Modular Polynomials,Top @anchor{index indices-and-tables}@anchor{d7} @unnumbered Indices and Tables @itemize * @item Index@footnote{../genindex.html} @item Module Index@footnote{../py-modindex.html} @item Search Page@footnote{../search.html} @end itemize @node Python Module Index,Index,Indices and Tables,Top @unnumbered Python Module Index @menu * sage.databases.conway: 0. * sage.databases.cremona: 1. * sage.databases.cunningham_tables: 2. * sage.databases.db_class_polynomials: 3. * sage.databases.db_modular_polynomials: 4. * sage.databases.findstat: 5. * sage.databases.jones: 6. * sage.databases.odlyzko: 7. * sage.databases.oeis: 8. * sage.databases.sloane: 9. * sage.databases.stein_watkins: a. * sage.databases.symbolic_data: b. @end menu @node Index,,Python Module Index,Top @unnumbered Index @printindex ge @c %**end of body @bye