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 type the following in Sage:
!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);
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 ticket #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.misc.misc import SAGE_SHARE
sage: os.path.isfile(os.path.join(SAGE_SHARE,'cremona','should_not_exist.db'))
False
Bases: sage.databases.cremona.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], 'a3': [[0, -1, 1, 0, 0], 0, 5],
'a2': [[0, -1, 1, -7820, -263580], 0, 1]}
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 , where E has rank r.
INPUT:
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]
Return the allgens table for conductor N.
INPUT:
OUTPUT:
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]]
Return the degphi table for conductor N.
INPUT:
OUTPUT:
EXAMPLES:
sage: c = CremonaDatabase()
sage: c.degphi(11) # optional - database_cremona_ellcurve
{'a1': 1}
sage: c.degphi(12001)['c1'] # optional - database_cremona_ellcurve
1640
Bases: sage.databases.sql_db.SQLDatabase
The Cremona database of elliptic curves.
EXAMPLES:
sage: c = CremonaDatabase()
sage: c.allcurves(11)
{'a1': [[0, -1, 1, -10, -20], 0, 5], 'a3': [[0, -1, 1, 0, 0], 0, 5],
'a2': [[0, -1, 1, -7820, -263580], 0, 1]}
Returns the allcurves table of curves of conductor N.
INPUT:
OUTPUT:
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]
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)
Returns the curves table of all optimal curves of conductor N.
INPUT:
OUTPUT:
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: list(sorted(CremonaDatabase().curves(990).keys()))
['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]
Return an elliptic curve with given label with some data about it from the database pre-filled in.
INPUT:
OUTPUT:
Note
For more details on LMFDB labels see 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
Returns the elliptic curve in the database of with minimal ainvs, if it exists, or raises a RuntimeError exception otherwise.
INPUT:
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 ticket #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
Returns the isogeny class of elliptic curves that are isogenous to the curve with given Cremona label.
INPUT:
OUTPUT:
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]
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]]]
Return an iterator through all curves in the database with given conductors.
INPUT:
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']
Return an iterator through all optimal curves in the database with given conductors.
INPUT:
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']
The largest conductor for which the database is complete.
OUTPUT:
EXAMPLES:
sage: c = CremonaDatabase('cremona mini')
sage: c.largest_conductor()
9999
Returns a list of all curves with given conductors.
INPUT:
OUTPUT:
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]
Returns a list of all optimal curves with given conductors.
INPUT:
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]
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:
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()
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:
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()
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
The smallest conductor for which the database is complete: always 1.
OUTPUT:
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
Build the CremonaDatabase with given name from scratch using the data_tgz tarball.
... note:
For data up to level 240000, this function takes about 3
minutes on a AMD Opteron(tm) Processor 6174. The resulting database
occupies 309MB disk space.
To create the large Cremona database from Cremona’s data_tgz tarball, run the following command:
sage: d = sage.databases.cremona.build('cremona','ecdata.tgz') # not tested
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'
Comparison function for curve id strings.
Note
Not the same as standard lexicographic order!
EXAMPLES:
sage: import sage.databases.cremona as cremona
sage: cremona.cmp_code('ba1','z1')
1
By contrast:
sage: cmp('ba1','z1')
-1
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:
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
Converts a Cremona label into an LMFDB label.
See parse_lmfdb_label() for an explanation of LMFDB labels.
INPUT:
OUTPUT:
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)
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:
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
Converts an LMFDB labe into a Cremona label.
See parse_lmfdb_label() for an explanation of LMFDB labels.
INPUT:
OUTPUT:
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'
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:
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'
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:
OUTPUT:
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
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:
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:
OUTPUT:
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)
Splits class+curve id string into its two parts.
EXAMPLES:
sage: import sage.databases.cremona as cremona
sage: cremona.split_code('ba2')
('ba', '2')