def egros_from_1728_over_K(K, S):
r"""
INPUT:
- ``K`` : a number field
- ``S`` : a list of primes of ``K``
OUTPUT:
A list with all elliptic curves over ``K`` with good reduction outside ``S`` and `j`-invariant equal to 1728.
REFERENCE:
J. E. Cremona and M. P. Lingham. Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes.
Experimental Mathematics, 16(3):303-312, 2007.
EXAMPLE::
sage:
"""
if K == QQ:
from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
return egros_from_j_1728(S)
Sprime = copy(S)
#we add suitable primes above 2
for p in K.primes_above(2):
if (K(2)).valuation(p) % 2 == 1 and p not in Sprime:
Sprime.append(p)
SprimenotS = [p for p in Sprime if p not in S]
selmergens, orders = K.selmer_group(Sprime, 4, orders=True)
curves = []
if len(SprimenotS) == 0:
for v in cartesian_product_iterator([xrange(b) for b in orders]):
w = prod([g**i for g, i in zip(selmergens, v)])
E = EllipticCurve([w, 0])
if has_good_reduction_outside_S_over_K(E, S):
curves.append(E)
return curves
for v in cartesian_product_iterator([xrange(b) for b in orders]):
w = prod([g**i for g, i in zip(selmergens, v)])
if len([1 for p in SprimenotS
if w.valuation(p) % 4 == 2]) == len(SprimenotS):
E = EllipticCurve([w, 0])
if has_good_reduction_outside_S_over_K(E, S):
curves.append(E)
return curves
def egros_from_1728_over_K(K,S):
r"""
INPUT:
- ``K`` : a number field
- ``S`` : a list of primes of ``K``
OUTPUT:
A list with all elliptic curves over ``K`` with good reduction outside ``S`` and `j`-invariant equal to 1728.
REFERENCE:
J. E. Cremona and M. P. Lingham. Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes.
Experimental Mathematics, 16(3):303-312, 2007.
EXAMPLE::
sage:
"""
if K == QQ:
from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
return egros_from_j_1728(S)
Sprime = copy(S)
#we add suitable primes above 2
for p in K.primes_above(2):
if (K(2)).valuation(p)%2 == 1 and p not in Sprime:
Sprime.append(p)
SprimenotS = [p for p in Sprime if p not in S]
selmergens,orders = K.selmer_group(Sprime,4,orders=True)
curves = []
if len(SprimenotS) == 0:
for v in cartesian_product_iterator([xrange(b) for b in orders]):
w = prod([g**i for g,i in zip(selmergens,v)])
E = EllipticCurve([w,0])
if has_good_reduction_outside_S_over_K(E,S):
curves.append(E)
return curves
for v in cartesian_product_iterator([xrange(b) for b in orders]):
w = prod([g**i for g,i in zip(selmergens,v)])
if len([1 for p in SprimenotS if w.valuation(p)%4 == 2]) == len(SprimenotS):
E = EllipticCurve([w,0])
if has_good_reduction_outside_S_over_K(E,S):
curves.append(E)
return curves
def testegros(SQ):
from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
SK = K.primes_above(prod(SQ))
# print 'SK',SK
curvesK = [E.change_ring(QQ).minimal_model() for E in egros_from_1728_over_K(K, SK)]
curvesQ = [E for E in egros_from_j_1728(SQ)]
print "number over K", len(curvesK)
print "number over Q", len(curvesQ)
# return curvesK,curvesQ
if len(curvesK) != len(curvesQ):
raise ValueError("They have found different number of curves")
for E in curvesK:
if E not in curvesQ:
print "I find this not in both", E
return curvesK, curvesQ
def testegros(SQ):
from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
SK = K.primes_above(prod(SQ))
# print 'SK',SK
curvesK = [
E.change_ring(QQ).minimal_model()
for E in egros_from_1728_over_K(K, SK)
]
curvesQ = [E for E in egros_from_j_1728(SQ)]
print 'number over K', len(curvesK)
print 'number over Q', len(curvesQ)
# return curvesK,curvesQ
if len(curvesK) != len(curvesQ):
raise ValueError('They have found different number of curves')
for E in curvesK:
if E not in curvesQ:
print 'I find this not in both', E
return curvesK, curvesQ