def egros_from_j_0(S=[]):
r"""
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
"""
Elist=[]
if not 3 in S:
return Elist
no2 = not 2 in S
for ei in xmrange([2] + [6]*len(S)):
u = prod([p**e for p,e in zip([-1]+S,ei)],QQ(1))
if no2:
u*=16 ## make sure 12|val(D,2)
Eu = EllipticCurve([0,0,0,0,u]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
def egros_from_j_0(S=[]):
r"""
Given a list of primes S, returns a list of elliptic curves over Q
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 `Q` 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]))
432
"""
Elist = []
if not 3 in S:
return Elist
no2 = not 2 in S
for ei in xmrange([2] + [6] * len(S)):
u = prod([p**e for p, e in zip([-1] + S, ei)], QQ(1))
if no2:
u *= 16 ## make sure 12|val(D,2)
Eu = EllipticCurve([0, 0, 0, 0, u]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
def egros_from_j_1728(S=[]):
r"""
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']
"""
Elist=[]
no2 = not 2 in S
for ei in xmrange([2] + [4]*len(S)):
u = prod([p**e for p,e in zip([-1]+S,ei)],QQ(1))
if no2:
u*=4 ## make sure 12|val(D,2)
Eu = EllipticCurve([0,0,0,u,0]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
def egros_from_j_1728(S=[]):
r"""
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']
"""
Elist = []
no2 = not 2 in S
for ei in xmrange([2] + [4] * len(S)):
u = prod([p**e for p, e in zip([-1] + S, ei)], QQ(1))
if no2:
u *= 4 ## make sure 12|val(D,2)
Eu = EllipticCurve([0, 0, 0, u, 0]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
