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: [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']
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: [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']
AUTHORS:
John Cremona (6 April 2009): initial version (over \(\QQ\) only).
- sage.schemes.elliptic_curves.ell_egros.curve_key(E1)¶
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: 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))
- sage.schemes.elliptic_curves.ell_egros.egros_from_j(j, S=[])¶
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: 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']
- sage.schemes.elliptic_curves.ell_egros.egros_from_j_0(S=[])¶
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: 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
- sage.schemes.elliptic_curves.ell_egros.egros_from_j_1728(S=[])¶
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: 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']
- sage.schemes.elliptic_curves.ell_egros.egros_from_jlist(jlist, S=[])¶
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 listS
.EXAMPLES:
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)]
- sage.schemes.elliptic_curves.ell_egros.egros_get_j(S=[], proof=None, verbose=False)¶
Returns 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
–True
/False
(defaultTrue
): the MW basis for auxiliary curves will be computed with this proof flag.verbose
–True
/False
(defaultFalse``
): ifTrue
, 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: 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
- sage.schemes.elliptic_curves.ell_egros.is_possible_j(j, S=[])¶
Tests 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: 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