Elliptic curves with prescribed good reduction¶

Construction of elliptic curves with good reduction outside a finite set of primes

A theorem of Shafarevich states that, over a number field \(K\), given any finite set \(S\) of primes of \(K\), there are (up to isomorphism) only a finite set of elliptic curves defined over \(K\) with good reduction at all primes outside \(S\). An explicit form of the theorem with an algorithm for finding this finite set was given in “Finding all elliptic curves with good reduction outside a given set of primes” by John Cremona and Mark Lingham, Experimental Mathematics 16 No.3 (2007), 303-312. The method requires computation of the class and unit groups of \(K\) as well as all the \(S\)-integral points on a collection of auxiliary elliptic curves defined over \(K\).

This implementation (April 2009) is only for the case \(K=\QQ\), where in many cases the determination of the necessary sets of \(S\)-integral points is possible. The main user-level function is EllipticCurves_with_good_reduction_outside_S(), defined in constructor.py. Users should note carefully the following points:

(1) the number of auxiliary curves to be considered is exponential in the size of \(S\) (specifically, \(2.6^s\) where \(s=|S|\)).

(2) For some of the auxiliary curves it is impossible at present to provably find all the \(S\)-integral points using the current algorithms, which rely on first finding a basis for their Mordell-Weil groups using 2-descent. A warning is output in cases where the set of points (and hence the final output) is not guaranteed to be complete. Using the proof=False flag suppresses these warnings.

EXAMPLES: We find all elliptic curves with good reduction outside 2, listing the label of each:

Sage

sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([2])]  # long time (5s on sage.math, 2013)
['32a1',
'32a2',
'32a3',
'32a4',
'64a1',
'64a2',
'64a3',
'64a4',
'128a1',
'128a2',
'128b1',
'128b2',
'128c1',
'128c2',
'128d1',
'128d2',
'256a1',
'256a2',
'256b1',
'256b2',
'256c1',
'256c2',
'256d1',
'256d2']

Python

>>> from sage.all import *
>>> [e.label() for e in EllipticCurves_with_good_reduction_outside_S([Integer(2)])]  # long time (5s on sage.math, 2013)
['32a1',
'32a2',
'32a3',
'32a4',
'64a1',
'64a2',
'64a3',
'64a4',
'128a1',
'128a2',
'128b1',
'128b2',
'128c1',
'128c2',
'128d1',
'128d2',
'256a1',
'256a2',
'256b1',
'256b2',
'256c1',
'256c2',
'256d1',
'256d2']

Sage Live

[e.label() for e in EllipticCurves_with_good_reduction_outside_S([2])]  # long time (5s on sage.math, 2013)

Secondly we try the same with \(S={11}\); note that warning messages are printed without proof=False (unless the optional database is installed: two of the auxiliary curves whose Mordell-Weil bases are required have conductors 13068 and 52272 so are in the database):

Sage

sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([11], proof=False)]  # long time (13s on sage.math, 2011)
['11a1', '11a2', '11a3', '121a1', '121a2', '121b1', '121b2', '121c1', '121c2', '121d1', '121d2', '121d3']

Python

>>> from sage.all import *
>>> [e.label() for e in EllipticCurves_with_good_reduction_outside_S([Integer(11)], proof=False)]  # long time (13s on sage.math, 2011)
['11a1', '11a2', '11a3', '121a1', '121a2', '121b1', '121b2', '121c1', '121c2', '121d1', '121d2', '121d3']

Sage Live

[e.label() for e in EllipticCurves_with_good_reduction_outside_S([11], proof=False)]  # long time (13s on sage.math, 2011)

AUTHORS:

John Cremona (6 April 2009): initial version (over \(\QQ\) only).

sage.schemes.elliptic_curves.ell_egros.curve_key(E1)[source]¶

Comparison key for elliptic curves over \(\QQ\).

The key is a tuple:

if the curve is in the database: (conductor, 0, label, number)
otherwise: (conductor, 1, a_invariants)

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.ell_egros import curve_key
sage: E = EllipticCurve_from_j(1728)
sage: curve_key(E)
(32, 0, 0, 2)
sage: E = EllipticCurve_from_j(1729)
sage: curve_key(E)
(2989441, 1, (1, 0, 0, -36, -1))

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_egros import curve_key
>>> E = EllipticCurve_from_j(Integer(1728))
>>> curve_key(E)
(32, 0, 0, 2)
>>> E = EllipticCurve_from_j(Integer(1729))
>>> curve_key(E)
(2989441, 1, (1, 0, 0, -36, -1))

Sage Live

from sage.schemes.elliptic_curves.ell_egros import curve_key
E = EllipticCurve_from_j(1728)
curve_key(E)
E = EllipticCurve_from_j(1729)
curve_key(E)

sage.schemes.elliptic_curves.ell_egros.egros_from_j(j, S=[])[source]¶

Given a rational j and a list of primes S, returns a list of elliptic curves over \(\QQ\) with j-invariant j and good reduction outside S, by checking all relevant quadratic twists.

INPUT:

j – a rational number
S – list of primes (default: empty list)

Note

Primality of elements of S is not checked, and the output is undefined if S is not a list or contains non-primes.

OUTPUT:

A sorted list of all elliptic curves defined over \(\QQ\) with \(j\)-invariant equal to \(j\) and with good reduction at all primes outside the list S.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.ell_egros import egros_from_j
sage: [e.label() for e in egros_from_j(0,[3])]
['27a1', '27a3', '243a1', '243a2', '243b1', '243b2']
sage: [e.label() for e in egros_from_j(1728,[2])]
['32a1', '32a2', '64a1', '64a4', '256b1', '256b2', '256c1', '256c2']
sage: elist=egros_from_j(-4096/11,[11])
sage: [e.label() for e in elist]
['11a3', '121d1']

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_egros import egros_from_j
>>> [e.label() for e in egros_from_j(Integer(0),[Integer(3)])]
['27a1', '27a3', '243a1', '243a2', '243b1', '243b2']
>>> [e.label() for e in egros_from_j(Integer(1728),[Integer(2)])]
['32a1', '32a2', '64a1', '64a4', '256b1', '256b2', '256c1', '256c2']
>>> elist=egros_from_j(-Integer(4096)/Integer(11),[Integer(11)])
>>> [e.label() for e in elist]
['11a3', '121d1']

Sage Live

from sage.schemes.elliptic_curves.ell_egros import egros_from_j
[e.label() for e in egros_from_j(0,[3])]
[e.label() for e in egros_from_j(1728,[2])]
elist=egros_from_j(-4096/11,[11])
[e.label() for e in elist]

sage.schemes.elliptic_curves.ell_egros.egros_from_j_0(S=[])[source]¶

Given a list of primes S, returns a list of elliptic curves over \(\QQ\) with j-invariant 0 and good reduction outside S, by checking all relevant sextic twists.

INPUT:

S – list of primes (default: empty list)

Note

Primality of elements of S is not checked, and the output is undefined if S is not a list or contains non-primes.

OUTPUT:

A sorted list of all elliptic curves defined over \(\QQ\) with \(j\)-invariant equal to \(0\) and with good reduction at all primes outside the list S.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.ell_egros import egros_from_j_0
sage: egros_from_j_0([])
[]
sage: egros_from_j_0([2])
[]
sage: [e.label() for e in egros_from_j_0([3])]
['27a1', '27a3', '243a1', '243a2', '243b1', '243b2']
sage: len(egros_from_j_0([2,3,5]))  # long time (8s on sage.math, 2013)
432

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_egros import egros_from_j_0
>>> egros_from_j_0([])
[]
>>> egros_from_j_0([Integer(2)])
[]
>>> [e.label() for e in egros_from_j_0([Integer(3)])]
['27a1', '27a3', '243a1', '243a2', '243b1', '243b2']
>>> len(egros_from_j_0([Integer(2),Integer(3),Integer(5)]))  # long time (8s on sage.math, 2013)
432

Sage Live

from sage.schemes.elliptic_curves.ell_egros import egros_from_j_0
egros_from_j_0([])
egros_from_j_0([2])
[e.label() for e in egros_from_j_0([3])]
len(egros_from_j_0([2,3,5]))  # long time (8s on sage.math, 2013)

sage.schemes.elliptic_curves.ell_egros.egros_from_j_1728(S=[])[source]¶

Given a list of primes S, returns a list of elliptic curves over \(\QQ\) with j-invariant 1728 and good reduction outside S, by checking all relevant quartic twists.

INPUT:

S – list of primes (default: empty list)

Note

Primality of elements of S is not checked, and the output is undefined if S is not a list or contains non-primes.

OUTPUT:

A sorted list of all elliptic curves defined over \(\QQ\) with \(j\)-invariant equal to \(1728\) and with good reduction at all primes outside the list S.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
sage: egros_from_j_1728([])
[]
sage: egros_from_j_1728([3])
[]
sage: [e.cremona_label() for e in egros_from_j_1728([2])]
['32a1', '32a2', '64a1', '64a4', '256b1', '256b2', '256c1', '256c2']

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
>>> egros_from_j_1728([])
[]
>>> egros_from_j_1728([Integer(3)])
[]
>>> [e.cremona_label() for e in egros_from_j_1728([Integer(2)])]
['32a1', '32a2', '64a1', '64a4', '256b1', '256b2', '256c1', '256c2']

Sage Live

from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
egros_from_j_1728([])
egros_from_j_1728([3])
[e.cremona_label() for e in egros_from_j_1728([2])]

sage.schemes.elliptic_curves.ell_egros.egros_from_jlist(jlist, S=[])[source]¶

Given a list of rational j and a list of primes S, returns a list of elliptic curves over \(\QQ\) with j-invariant in the list and good reduction outside S.

INPUT:

j – list of rational numbers
S – list of primes (default: empty list)

Note

Primality of elements of S is not checked, and the output is undefined if S is not a list or contains non-primes.

OUTPUT:

A sorted list of all elliptic curves defined over \(\QQ\) with \(j\)-invariant in the list jlist and with good reduction at all primes outside the list S.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j, egros_from_jlist
sage: jlist=egros_get_j([3])
sage: elist=egros_from_jlist(jlist,[3])
sage: [e.label() for e in elist]
['27a1', '27a2', '27a3', '27a4', '243a1', '243a2', '243b1', '243b2']
sage: [e.ainvs() for e in elist]
[(0, 0, 1, 0, -7),
(0, 0, 1, -270, -1708),
(0, 0, 1, 0, 0),
(0, 0, 1, -30, 63),
(0, 0, 1, 0, -1),
(0, 0, 1, 0, 20),
(0, 0, 1, 0, 2),
(0, 0, 1, 0, -61)]

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_egros import egros_get_j, egros_from_jlist
>>> jlist=egros_get_j([Integer(3)])
>>> elist=egros_from_jlist(jlist,[Integer(3)])
>>> [e.label() for e in elist]
['27a1', '27a2', '27a3', '27a4', '243a1', '243a2', '243b1', '243b2']
>>> [e.ainvs() for e in elist]
[(0, 0, 1, 0, -7),
(0, 0, 1, -270, -1708),
(0, 0, 1, 0, 0),
(0, 0, 1, -30, 63),
(0, 0, 1, 0, -1),
(0, 0, 1, 0, 20),
(0, 0, 1, 0, 2),
(0, 0, 1, 0, -61)]

Sage Live

from sage.schemes.elliptic_curves.ell_egros import egros_get_j, egros_from_jlist
jlist=egros_get_j([3])
elist=egros_from_jlist(jlist,[3])
[e.label() for e in elist]
[e.ainvs() for e in elist]

sage.schemes.elliptic_curves.ell_egros.egros_get_j(S=[], proof=None, verbose=False)[source]¶

Return a list of rational \(j\) such that all elliptic curves defined over \(\QQ\) with good reduction outside \(S\) have \(j\)-invariant in the list, sorted by height.

INPUT:

S – list of primes (default: empty list)
proof – boolean (default: True); the MW basis for auxiliary curves will be computed with this proof flag
verbose – boolean (default: False); if True, some details of the computation will be output

Note

Proof flag: The algorithm used requires determining all S-integral points on several auxiliary curves, which in turn requires the computation of their generators. This is not always possible (even in theory) using current knowledge.

The value of this flag is passed to the function which computes generators of various auxiliary elliptic curves, in order to find their S-integral points. Set to False if the default (True) causes warning messages, but note that you can then not rely on the set of invariants returned being complete.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j
sage: egros_get_j([])
[1728]
sage: egros_get_j([2])  # long time (3s on sage.math, 2013)
[128, 432, -864, 1728, 3375/2, -3456, 6912, 8000, 10976, -35937/4, 287496, -784446336, -189613868625/128]
sage: egros_get_j([3])  # long time (3s on sage.math, 2013)
[0, -576, 1536, 1728, -5184, -13824, 21952/9, -41472, 140608/3, -12288000]
sage: jlist=egros_get_j([2,3]); len(jlist) # long time (30s)
83

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_egros import egros_get_j
>>> egros_get_j([])
[1728]
>>> egros_get_j([Integer(2)])  # long time (3s on sage.math, 2013)
[128, 432, -864, 1728, 3375/2, -3456, 6912, 8000, 10976, -35937/4, 287496, -784446336, -189613868625/128]
>>> egros_get_j([Integer(3)])  # long time (3s on sage.math, 2013)
[0, -576, 1536, 1728, -5184, -13824, 21952/9, -41472, 140608/3, -12288000]
>>> jlist=egros_get_j([Integer(2),Integer(3)]); len(jlist) # long time (30s)
83

Sage Live

from sage.schemes.elliptic_curves.ell_egros import egros_get_j
egros_get_j([])
egros_get_j([2])  # long time (3s on sage.math, 2013)
egros_get_j([3])  # long time (3s on sage.math, 2013)
jlist=egros_get_j([2,3]); len(jlist) # long time (30s)

sage.schemes.elliptic_curves.ell_egros.is_possible_j(j, S=[])[source]¶

Test if the rational \(j\) is a possible \(j\)-invariant of an elliptic curve with good reduction outside \(S\).

Note

The condition used is necessary but not sufficient unless S contains both 2 and 3.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.ell_egros import is_possible_j
sage: is_possible_j(0,[])
False
sage: is_possible_j(1728,[])
True
sage: is_possible_j(-4096/11,[11])
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_egros import is_possible_j
>>> is_possible_j(Integer(0),[])
False
>>> is_possible_j(Integer(1728),[])
True
>>> is_possible_j(-Integer(4096)/Integer(11),[Integer(11)])
True

Sage Live

from sage.schemes.elliptic_curves.ell_egros import is_possible_j
is_possible_j(0,[])
is_possible_j(1728,[])
is_possible_j(-4096/11,[11])