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""" Cremona's tables of elliptic curves
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::
sage -i database_cremona_ellcurve
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::
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);
while the full version has the layout::
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); """ #***************************************************************************** # Copyright (C) 2014 John Cremona <john.cremona@gmail.com> # Copyright (C) 2011 R. Andrew Ohana <andrew.ohana@gmail.com> # Copyright (C) 2005 William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from __future__ import print_function
from sage.misc.prandom import randint
import sage.schemes.elliptic_curves.constructor as elliptic from sage.misc.package import is_package_installed from sage.misc.all import walltime
import re import string
_cremonaSkeleton = { 't_class': { 'conductor': {'sql':'INTEGER', 'index':True}, 'class': {'sql':'TEXT', 'primary_key':True}, 'rank': {'sql':'INTEGER'}, 'L': {'sql':'REAL'}, 'deg': {'sql':'INTEGER'} }, 't_curve': { 'class': {'sql':'TEXT', 'index':True}, 'curve': {'sql':'TEXT', 'primary_key':True}, 'eqn': {'sql':'TEXT', 'unique':True}, 'gens': {'sql':'TEXT'}, 'tors': {'sql':'INTEGER'}, 'cp': {'sql':'INTEGER'}, 'om': {'sql':'REAL'}, 'reg': {'sql':'REAL'}, 'sha': {'sql':'NOTYPE'} } } _miniCremonaSkeleton = { 't_class': { 'conductor': {'sql':'INTEGER', 'index':True}, 'class': {'sql':'TEXT', 'primary_key':True}, 'rank': {'sql':'INTEGER'} }, 't_curve': { 'class': {'sql':'TEXT', 'index':True}, 'curve': {'sql':'TEXT', 'primary_key':True}, 'eqn': {'sql':'TEXT', 'unique':True}, 'tors': {'sql':'INTEGER'} } }
for t in _cremonaSkeleton: for c in _cremonaSkeleton[t]: _cremonaSkeleton[t][c] = verify_column(_cremonaSkeleton[t][c]) for c in _miniCremonaSkeleton[t]: _miniCremonaSkeleton[t][c] = verify_column(_miniCremonaSkeleton[t][c])
def build(name, data_tgz, largest_conductor=0, mini=False, decompress=True): """ Build the CremonaDatabase with given name from scratch using the data_tgz tarball.
.. note::
For data up to level 350000, this function takes about 3m40s. The resulting database occupies 426MB disk space.
To create the large Cremona database from Cremona's data_tgz tarball, obtainable from http://homepages.warwick.ac.uk/staff/J.E.Cremona/ftp/data/, run the following command::
sage: d = sage.databases.cremona.build('cremona','ecdata.tgz') # not tested """ db_path = os.path.join(SAGE_SHARE,'cremona',name.replace(' ','_')+'.db') if os.path.exists(db_path): raise RuntimeError('Please (re)move %s before building '%db_path \ + 'database') if not os.path.exists(data_tgz): raise IOError("The data file is not at %s"%data_tgz) t = walltime()
if decompress: cmd = "tar zxvf %s"%data_tgz n = os.system(cmd) if n: raise RuntimeError("Error extracting tarball.") if mini: c = MiniCremonaDatabase(name,False,True) else: c = LargeCremonaDatabase(name,False,True) # The following line assumes that the tarball extracts to a # directory called 'ecdata' c._init_from_ftpdata('ecdata', largest_conductor) print("Total time: ", walltime(t))
def 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.
.. note::
990h3 is the optimal curve in that class, so doesn't obey this rule.
INPUT:
- ``id`` - str of form letter code followed by an integer, e.g., a3, bb5, etc.
OUTPUT: bool
EXAMPLES::
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 """
def 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.
.. 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.
INPUT:
- ``n`` (int) -- a non-negative integer
OUTPUT: str
EXAMPLES::
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'
TESTS::
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 """ else:
def old_cremona_letter_code(n): r""" Returns the *old* Cremona letter code corresponding to an integer. integer.
For example::
1 --> A 26 --> Z 27 --> AA 52 --> ZZ 53 --> AAA etc.
INPUT:
- ``n`` - int
OUTPUT: str
EXAMPLES::
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' """
old_cremona_label_regex = re.compile(r'(\d+)([A-Z]*)(\d*)$') cremona_label_regex = re.compile(r'(\d+)([a-z]*)(\d*)$') lmfdb_label_regex = re.compile(r'(\d+)\.([a-z]+)(\d*)$')
def 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:
- ``label`` - str
OUTPUT:
- ``int`` - the conductor - ``str`` - the isogeny class label - ``int`` - the number
EXAMPLES::
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)
Valid old Cremona labels are allowed::
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
TESTS::
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 """
# convert old cremona labels to new ones
# verify cremona label is valid
def 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:
- Isogeny classes with the same conductor are ordered lexicographically by the coefficients in the q-expansion of the associated modular form.
- Curves within the same isogeny class are ordered lexicographically by the a-invariants of the minimal model.
The format is <conductor>.<iso><curve>, 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:
- ``label`` - str
OUTPUT:
- ``int`` - the conductor - ``str`` - the isogeny class label - ``int`` - the number
EXAMPLES::
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) """ raise ValueError(label + " is not a valid LMFDB label") iso = "a"
def split_code(key): """ Splits class+curve id string into its two parts.
EXAMPLES::
sage: import sage.databases.cremona as cremona sage: cremona.split_code('ba2') ('ba', '2') """
def class_to_int(k): """ Converts class id string into an integer. Note that this is the inverse of cremona_letter_code.
EXAMPLES::
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' """
def sort_key(key1): """ Comparison key for curve id strings.
.. note::
Not the same as standard lexicographic order!
EXAMPLES::
sage: from sage.databases.cremona import sort_key sage: l = ['ba1', 'z1'] sage: sorted(l, key=sort_key) ['z1', 'ba1'] """
def cremona_to_lmfdb(cremona_label, CDB=None): """ Converts a Cremona label into an LMFDB label.
See :func:`parse_lmfdb_label` for an explanation of LMFDB labels.
INPUT:
- ``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') - ``CDB`` -- the Cremona database in which to look up the isogeny classes of the same conductor.
OUTPUT:
- ``lmfdb_label`` -- a string, the corresponding LMFDB label.
EXAMPLES::
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'
TESTS::
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) """ raise ValueError("Invalid Cremona label") else:
def lmfdb_to_cremona(lmfdb_label, CDB=None): """ Converts an LMFDB labe into a Cremona label.
See :func:`parse_lmfdb_label` for an explanation of LMFDB labels.
INPUT:
- ``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') - ``CDB`` -- the Cremona database in which to look up the isogeny classes of the same conductor.
OUTPUT:
- ``cremona_label`` -- a string, the corresponding Cremona label.
EXAMPLES::
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' """ raise ValueError("Invalid LMFDB label") else:
class MiniCremonaDatabase(SQLDatabase): """ The Cremona database of elliptic curves.
EXAMPLES::
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]} """ def __init__(self, name, read_only=True, build=False): """ Initialize the database.
TESTS::
sage: c = CremonaDatabase('cremona mini') sage: c.name 'cremona mini' """ if name is None: raise RuntimeError('The database must have a name.') if read_only: raise RuntimeError('The database must not be read_only.') SQLDatabase.__init__(self, db_path, read_only=read_only, \ skeleton=_miniCremonaSkeleton) return + "exist") raise RuntimeError('Database at %s does '%(self.__dblocation__) \ + 'not appear to be a valid SQL Cremona database.')
def __iter__(self): """ Returns an iterator through all EllipticCurve objects in the Cremona database.
TESTS::
sage: it = CremonaDatabase().__iter__() sage: next(it).label() '11a1' sage: next(it).label() '11a2' sage: next(it).label() '11a3' sage: next(it).label() '14a1' sage: skip = [next(it) for _ in range(100)] sage: next(it).label() '45a3' """
def __getitem__(self, N): """ If N is an integer, return all data about level N in the database. If N is a string it must be a Cremona label, in which case return the corresponding elliptic curve, if it is in the database.
INPUT:
- ``N`` - int or str
OUTPUT: dict (if N is an int) or EllipticCurve (if N is a str)
TESTS::
sage: c = CremonaDatabase() sage: c[11]['allcurves']['a2'] [[0, -1, 1, -7820, -263580], 0, 1] sage: c['11a2'] Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field """
except ValueError: raise KeyError("N (=%s) must be a string or positive integer."%N)
raise KeyError("N (=%s) must be a string or positive integer."%N)
ret['allbsd'] = self.allbsd(N) ret['degphi'] = self.degphi(N) ret['allgens'] = self.allgens(N)
def __repr__(self): """ String representation of this database.
TESTS::
sage: c = CremonaDatabase('cremona mini') sage: c.__repr__() "Cremona's database of elliptic curves with conductor at most 9999" """ + str(self.largest_conductor())
def allcurves(self, N): """ Returns the allcurves table of curves of conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:[ainvs, rank, tor], ...
EXAMPLES::
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] """ + 'rank,tors FROM t_curve,t_class USING(class) WHERE ' \ + 'conductor=?',(int(N),)):
def curves(self, N): """ Returns the curves table of all *optimal* curves of conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:[ainvs, rank, tor], ...
EXAMPLES:
Optimal curves of conductor 37::
sage: CremonaDatabase().curves(37) {'a1': [[0, 0, 1, -1, 0], 1, 1], 'b1': [[0, 1, 1, -23, -50], 0, 3]}
Note the 'h3', which is the unique case in the tables where the optimal curve doesn't have label ending in 1::
sage: sorted(CremonaDatabase().curves(990)) ['a1', 'b1', 'c1', 'd1', 'e1', 'f1', 'g1', 'h3', 'i1', 'j1', 'k1', 'l1']
TESTS::
sage: c = CremonaDatabase() sage: c.curves(12001)['a1'] # optional - database_cremona_ellcurve [[1, 0, 0, -101, 382], 1, 1] """ + 'rank,tors FROM t_curve,t_class USING(class) WHERE ' \ + 'curve=class||1 AND conductor=?',(int(N),)):
def coefficients_and_data(self, label): """ Return the Weierstrass coefficients and other data for the curve with given label.
EXAMPLES::
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
Check that :trac:`17904` is fixed::
sage: 'gens' in CremonaDatabase().coefficients_and_data('100467a2')[1] # optional - database_cremona_ellcurve True
""" # There are two possible strings: the Cremona label and the LMFDB label. # They are distinguished by the presence of a period. else:
+ 'FROM t_curve,t_class USING(class) WHERE curve=?', (label,)) else: q = self.__connection__.cursor().execute("SELECT eqn,rank,tors," \ + "deg,gens,cp,om,L,reg,sha FROM t_curve,t_class " \ + "USING(class) WHERE curve=?",(label,)) + " in the database" elif is_package_installed('database_cremona_ellcurve'): message = "There is no elliptic curve with label " + label \ + " in the currently available databases" else: message = "There is no elliptic curve with label " \ + label + " in the default database; try installing " \ + "the optional package database_cremona_ellcurve which " \ + "contains the complete Cremona database" 'rank': c[1], 'torsion_order': c[2], 'conductor': N} data['modular_degree'] = (c[3]) data['gens'] = eval(c[4]) data['db_extra'] = list(c[5:])
def data_from_coefficients(self, ainvs): """ Return elliptic curve data for the curve with given Weierstrass coefficients.
EXAMPLES::
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
Check that :trac:`17904` is fixed::
sage: ai = EllipticCurve('100467a2').ainvs() # optional - database_cremona_ellcurve sage: 'gens' in CremonaDatabase().data_from_coefficients(ai) # optional - database_cremona_ellcurve True """ + 'FROM t_curve,t_class USING(class) WHERE eqn=?', (ainvs.replace(' ', ''),)) else: q = self.__connection__.cursor().execute("SELECT curve,rank,tors," + "deg,gens,cp,om,L,reg,sha FROM t_curve,t_class " + "USING(class) WHERE eqn=?", (ainvs.replace(' ', ''),)) + ainvs + " in the database") 'rank': c[1], 'torsion_order': c[2], 'conductor': N} data['modular_degree'] = (c[3]) data['gens'] = eval(c[4]) data['db_extra'] = list(c[5:])
def elliptic_curve_from_ainvs(self, ainvs): """ Returns the elliptic curve in the database of with minimal ainvs, if it exists, or raises a RuntimeError exception otherwise.
INPUT:
- ``ainvs`` - list (5-tuple of int's); the minimal Weierstrass model for an elliptic curve
OUTPUT: EllipticCurve
EXAMPLES::
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
Old (pre-2006) Cremona labels are also allowed::
sage: c.elliptic_curve('9450KKKK1') Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 5*x + 7 over Rational Field
Make sure :trac:`12565` is fixed::
sage: c.elliptic_curve('10a1') Traceback (most recent call last): ... ValueError: There is no elliptic curve with label 10a1 in the database """
def elliptic_curve(self, label): """ Return an elliptic curve with given label with some data about it from the database pre-filled in.
INPUT:
- ``label`` - str (Cremona or LMFDB label)
OUTPUT:
- an :class:`sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field`
.. note::
For more details on LMFDB labels see :func:`parse_lmfdb_label`.
EXAMPLES::
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
You can also use LMFDB labels::
sage: c.elliptic_curve('462.f3') Elliptic Curve defined by y^2 + x*y = x^3 - 363*x + 1305 over Rational Field """
def iter(self, conductors): """ Return an iterator through all curves in the database with given conductors.
INPUT:
- ``conductors`` - list or generator of ints
OUTPUT: generator that iterates over EllipticCurve objects.
EXAMPLES::
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'] """ + 'FROM t_curve,t_class USING(class) WHERE conductor=?', \ (int(N),)):
def isogeny_classes(self, 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::
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]]] """ # test if not first curve in class else:
def isogeny_class(self, label): """ Returns the isogeny class of elliptic curves that are isogenous to the curve with given Cremona label.
INPUT:
- ``label`` - string
OUTPUT:
- ``list`` - list of EllipticCurve objects.
EXAMPLES::
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] """ + "WHERE class=?",(str(conductor)+iso,))
def iter_optimal(self, conductors): """ Return an iterator through all optimal curves in the database with given conductors.
INPUT:
- ``conductors`` - list or generator of ints
OUTPUT:
generator that iterates over EllipticCurve objects.
EXAMPLES:
We list optimal curves with conductor up to 20::
sage: [e.cremona_label() for e in CremonaDatabase().iter_optimal([11..20])] ['11a1', '14a1', '15a1', '17a1', '19a1', '20a1']
Note the unfortunate 990h3 special case::
sage: [e.cremona_label() for e in CremonaDatabase().iter_optimal([990])] ['990a1', '990b1', '990c1', '990d1', '990e1', '990f1', '990g1', '990h3', '990i1', '990j1', '990k1', '990l1'] """ + 'FROM t_class WHERE conductor=990'): else: + 'FROM t_curve,t_class USING(class) WHERE curve=class||1 ' \ + 'AND conductor=?',(int(N),)):
def list(self, conductors): """ Returns a list of all curves with given conductors.
INPUT:
- ``conductors`` - list or generator of ints
OUTPUT:
- list of EllipticCurve objects.
EXAMPLES::
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] """
def list_optimal(self, conductors): """ Returns a list of all optimal curves with given conductors.
INPUT:
- ``conductors`` - list or generator of ints list of EllipticCurve objects.
OUTPUT:
list of EllipticCurve objects.
EXAMPLES::
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] """
def largest_conductor(self): """ The largest conductor for which the database is complete.
OUTPUT:
- ``int`` - largest conductor
EXAMPLES::
sage: c = CremonaDatabase('cremona mini') sage: c.largest_conductor() 9999 """ #print "Computing largest conductor." + 't_class ORDER BY conductor DESC LIMIT 1')
def smallest_conductor(self): """ The smallest conductor for which the database is complete: always 1.
OUTPUT:
- ``int`` - smallest conductor
.. 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.
EXAMPLES::
sage: CremonaDatabase().smallest_conductor() 1 """
def conductor_range(self): """ 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::
sage: c = CremonaDatabase('cremona mini') sage: c.conductor_range() (1, 10000) """
def number_of_curves(self, 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:
- ``N`` - int - ``i`` - int or str
OUTPUT: int
EXAMPLES::
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() """ return self.__number_of_curves__ + 'FROM t_curve') + 'FROM t_curve,t_class USING(class) WHERE conductor=?', \ (int(N),)) i = cremona_letter_code(i) + 't_curve WHERE class=?',(str(N)+i,))
def number_of_isogeny_classes(self, 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:
- ``N`` - int
OUTPUT: int
EXAMPLES::
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() """ return self.__number_of_isogeny_classes__ + 'FROM t_class') + 't_class WHERE conductor=?',(int(N),))
def random(self): """ Returns a random curve from the database.
EXAMPLES::
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 """ + 't_class WHERE conductor>=? ORDER BY conductor',(int(N),)) except StopIteration: N = 11
############################################################################### # Functions for loading data from Cremona's ftpdata directory. ############################################################################### def _init_from_ftpdata(self, ftpdata, largest_conductor=0): """ Create the SQL Cremona Database from the Cremona data directory, which is available from Cremona's website. I.e., just wget Cremona's database to a local directory.
To create the large database from Cremona's text files, see sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.MiniCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested sage: d._init_from_ftpdata('.') # not tested """ if self.__read_only__: raise RuntimeError("The database must not be read_only.")
if not os.path.exists(ftpdata): raise RuntimeError("The cremona ftpdata directory '" + ftpdata \ + "' does not exist.")
if largest_conductor: print("largest conductor =", largest_conductor) self.__largest_conductor__ = largest_conductor
# Since July 2014 the data files have been arranged in # subdirectories (see trac #16903). allcurves_dir = os.path.join(ftpdata,'allcurves') allbsd_dir = os.path.join(ftpdata,'allbsd') allgens_dir = os.path.join(ftpdata,'allgens') degphi_dir = os.path.join(ftpdata,'degphi') num_curves, num_iso_classes = self._init_allcurves(allcurves_dir, largest_conductor) self.__number_of_curves__ = num_curves self.__number_of_isogeny_classes__ = num_iso_classes if hasattr(self, 'degphi'): self._init_degphi(degphi_dir, largest_conductor) if hasattr(self, 'allbsd'): self._init_allbsd(allbsd_dir, largest_conductor) if hasattr(self, 'allgens'): self._init_allgens(allgens_dir, largest_conductor) self.vacuum()
def _init_allcurves(self, ftpdata, largest_conductor=0): """ Initialize the allcurves table by reading the corresponding ftpdata files and importing them into the database.
To create the large database from Cremona's text files, see sage.databases.cremona.build, do NOT run this method directly.
INPUT:
- `ftpdata` (string) -- the name of the directory in which the data is
- ``largest_conductor`` - int (default: 0), if 0, then only include data up to that conductor.
OUTPUT:
- ``int`` - number_of_curves - ``int`` - number_of_isogeny_classes
EXAMPLES::
sage: d = sage.databases.cremona.MiniCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested sage: d._init_allcurves('.', 11) # not tested (3, 1) """ if self.__read_only__: raise RuntimeError("The database must not be read_only.") files = sorted(os.listdir(ftpdata)) name = 'allcurves' num_curves = 0 num_iso_classes = 0 con = self.get_connection() for F in files: if not F[:len(name)] == name: continue print("Inserting", F) class_data = [] curve_data = [] for L in open(ftpdata + "/" + F).readlines(): N, iso, num, ainvs, r, tor = L.split() if largest_conductor and int(N) > largest_conductor: break cls = N+iso cur = cls+num if num == "1": class_data.append((N,cls,r)) num_iso_classes += 1 curve_data.append((cur,cls,ainvs,tor)) num_curves += 1 con.executemany('INSERT INTO t_class (conductor,class,rank) ' \ + 'VALUES (?,?,?)', class_data) con.executemany('INSERT INTO t_curve (curve,class,eqn,tors) ' \ + 'VALUES (?,?,?,?)', curve_data) print("Committing...") print("num_iso_classes =", num_iso_classes) self.commit() if largest_conductor and int(N) > largest_conductor: break return num_curves, num_iso_classes
class LargeCremonaDatabase(MiniCremonaDatabase): """ The Cremona database of elliptic curves.
EXAMPLES::
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]} """ def __init__(self, name, read_only=True, build=False): """ Initialize the database.
TESTS::
sage: c = CremonaDatabase('cremona') # optional - database_cremona_ellcurve sage: c.name # optional - database_cremona_ellcurve 'cremona' """ if name is None: raise RuntimeError('The database must have a name.') if read_only: raise RuntimeError('The database must not be read_only.') SQLDatabase.__init__(self, db_path, read_only=read_only, \ skeleton=_cremonaSkeleton) return + "exist") SQLDatabase.__init__(self, db_path, read_only=read_only) if self.get_skeleton() != _cremonaSkeleton: raise RuntimeError('Database at %s does '%(self.__dblocation__) \ + 'not appear to be a valid SQL Cremona database.')
def allbsd(self, N): r""" Return the allbsd table for conductor N. The entries are::
[id, tamagawa_product, Omega_E, L, Reg_E, Sha_an(E)]
where id is the isogeny class (letter) followed by a number, e.g., b3, and L is `L^r(E,1)/r!`, where E has rank r.
INPUT:
- ``N`` - int, the conductor
OUTPUT: dict containing the allbsd table for each isogeny class in conductor N
EXAMPLES::
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] """ ret = {} for c in self.__connection__.cursor().execute('SELECT curve,cp,om,L,' \ + 'reg,sha FROM t_curve,t_class USING(class) WHERE conductor=?', \ (int(N),)): N,iso,num = parse_cremona_label(c[0]) ret[iso+str(num)] = list(c[1:]) return ret
def allgens(self, N): """ Return the allgens table for conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:[points, ...], ...
EXAMPLES::
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]] """ ret = {} for c in self.__connection__.cursor().execute('SELECT curve,gens ' \ + 'FROM t_curve,t_class USING(class) WHERE conductor=?',(int(N),)): N,iso,num = parse_cremona_label(c[0]) ret[iso+str(num)] = eval(c[1]) return ret
def degphi(self, N): """ Return the degphi table for conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:degphi, ...
EXAMPLES::
sage: c = CremonaDatabase() sage: c.degphi(11) # optional - database_cremona_ellcurve {'a1': 1} sage: c.degphi(12001)['c1'] # optional - database_cremona_ellcurve 1640 """ ret = {} for c in self.__connection__.cursor().execute('SELECT curve,deg FROM' \ + ' t_curve,t_class USING(class) WHERE curve=class||1 AND ' \ + 'conductor=?', (int(N),)): N,iso,num = parse_cremona_label(c[0]) ret[iso+str(num)] = c[1] return ret
def _init_degphi(self, ftpdata, largest_conductor=0): """ Initialize the degphi table by reading the corresponding ftpdata files and importing them into the database.
To create the large database from Cremona's text files, see sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.LargeCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested sage: d._init_degphi('.') # not tested """ if self.__read_only__: raise RuntimeError("The database must not be read_only.") files = sorted(os.listdir(ftpdata)) name = "degphi" con = self.get_connection() for F in files: if not F[:len(name)] == name: continue print("Inserting", F) class_data = [] for L in open(ftpdata + "/" + F).readlines(): N, iso, num, degree, primes, curve = L.split() if largest_conductor and int(N) > largest_conductor: break class_data.append((degree,N+iso)) con.executemany('UPDATE t_class SET deg=? WHERE class=?', \ class_data) print("Committing...") self.commit() if largest_conductor and int(N) > largest_conductor: break
def _init_allbsd(self, ftpdata, largest_conductor=0): """ Initialize the allbsd table by reading the corresponding ftpdata files and importing them into the database.
To create the large database from Cremona's text files, see sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.LargeCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested sage: d._init_allbsd('.') # not tested """ if self.__read_only__: raise RuntimeError("The database must not be read_only.") files = sorted(os.listdir(ftpdata)) name = "allbsd" con = self.get_connection() for F in files: if not F[:len(name)] == name: continue print("Inserting", F) curve_data = [] class_data = [] for L in open(ftpdata + "/" + F).readlines(): N, iso, num, eqn, rank, tor, cp, om, L, reg, sha = L.split() if largest_conductor and int(N) > largest_conductor: break cls = N+iso if num == "1": class_data.append((L,cls)) curve_data.append((cp,om,reg,eval(sha),cls+num)) con.executemany("UPDATE t_class SET L=? WHERE class=?", class_data) con.executemany("UPDATE t_curve SET cp=?,om=?,reg=?,sha=? WHERE " \ + "curve=?", curve_data) print("Committing...") self.commit() if largest_conductor and int(N) > largest_conductor: break
def _init_allgens(self, ftpdata, largest_conductor=0): """ Initialize the allgens table by reading the corresponding ftpdata files and importing them into the database.
To create the large database from Cremona's text files, see sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.LargeCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested sage: d._init_allgens('.') # not tested """ if self.__read_only__: raise RuntimeError("The database must not be read_only.") files = sorted(os.listdir(ftpdata)) name = "allgens" con = self.get_connection() for F in files: if not F[:len(name)] == name: continue print("Inserting", F) curve_data = [] for L in open(ftpdata + "/" + F).readlines(): v = L.split() if largest_conductor and int(v[0]) > largest_conductor: break gens = '['+','.join(v[6:6+int(v[4])]).replace(':',',')+']' curve_data.append((gens,''.join(v[:3]))) con.executemany("UPDATE t_curve SET gens=? WHERE curve=?", \ curve_data) print("Committing...") if largest_conductor and int(v[0]) > largest_conductor: break
_db = None def CremonaDatabase(name=None,mini=None,set_global=None): """ Initializes the Cremona database with name ``name``. If ``name`` is ``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 ``mini=True``, otherwise it must be set to ``False``.
If you would like other components of Sage to use this database, mark ``set_global=True``.
TESTS::
sage: c = CremonaDatabase() sage: isinstance(c, sage.databases.cremona.MiniCremonaDatabase) True sage: isinstance(c, sage.databases.cremona.LargeCremonaDatabase) # optional - database_cremona_ellcurve True
Verify that :trac:`12341` has been resolved::
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 """ global _db name = 'cremona' else: mini = False raise ValueError('mini must be set as either True or False') else: _db = LargeCremonaDatabase(name) |