def aplist(E, B=100):
"""
Compute aplist for an elliptic curve E over Q(sqrt(5)), as a
string->number dictionary.
INPUT:
- E -- an elliptic curve
- B -- a positive integer (default: 100)
OUTPUT:
- dictionary mapping strings (labeled primes) to Python ints,
with keys the primes of P with norm up to B such that the
norm of the conductor is coprime to the characteristic of P.
"""
from psage.modform.hilbert.sqrt5.tables import canonical_gen
v = {}
from psage.modform.hilbert.sqrt5.sqrt5 import F
labels, primes = labeled_primes_of_bounded_norm(F, B)
from sage.all import ZZ
N = E.conductor()
try:
N = ZZ(N.norm())
except:
N = ZZ(N)
for i in range(len(primes)):
p = primes[i]
k = p.residue_field()
if N.gcd(k.cardinality()) == 1:
v[labels[i]] = int(k.cardinality() + 1 -
E.change_ring(k).cardinality())
return v
def __init__(self, K, F, minimal_ramification=1):
minimal_ramification = ZZ(minimal_ramification)
# print "entering WeakPadicGaloisExtension with"
# print "F = %s"%F
# if F is a polynomial, replace it by the list of its irreducible factors
# if isinstance(F, Polynomial):
# F = [f for f, m in F.factor()]
assert K.is_Qp(), "for the moment, K has to be Q_p"
# assert not K.p().divides(minimal_ramification), "minimal_ramification has to be prime to p"
if not isinstance(F, Polynomial):
if F == []:
F = PolynomialRing(K.number_field(), 'x')(1)
else:
F = prod(F)
self._base_field = K
L = K.weak_splitting_field(F)
e = ZZ(L.absolute_ramification_degree() /
K.absolute_ramification_degree())
if not minimal_ramification.divides(e):
# enlarge the absolute ramification index of vL
# such that minimal_ramification divides e(vL/vK):
m = ZZ(minimal_ramification / e.gcd(minimal_ramification))
# assert not self.p().divides(m), "p = %s, m = %s, e = %s,\nminimal_ramification = %s"%(self.p(), m, e, minimal_ramification)
L = L.ramified_extension(m)
# if m was not prime to p, L/K may not be weak Galois anymore
else:
m = ZZ(1)
self._extension_field = L
self._ramification_degree = e * m
self._degree = ZZ(L.absolute_degree() / K.absolute_degree())
self._inertia_degree = ZZ(L.absolute_inertia_degree() /
K.absolute_inertia_degree())
assert self._degree == self._ramification_degree * self._inertia_degree
def aplist(E, B=100):
"""
Compute aplist for an elliptic curve E over Q(sqrt(5)), as a
string->number dictionary.
INPUT:
- E -- an elliptic curve
- B -- a positive integer (default: 100)
OUTPUT:
- dictionary mapping strings (labeled primes) to Python ints,
with keys the primes of P with norm up to B such that the
norm of the conductor is coprime to the characteristic of P.
"""
from psage.modform.hilbert.sqrt5.tables import canonical_gen
v = {}
from psage.modform.hilbert.sqrt5.sqrt5 import F
labels, primes = labeled_primes_of_bounded_norm(F, B)
from sage.all import ZZ
N = E.conductor()
try:
N = ZZ(N.norm())
except:
N = ZZ(N)
for i in range(len(primes)):
p = primes[i]
k = p.residue_field()
if N.gcd(k.cardinality()) == 1:
v[labels[i]] = int(k.cardinality() + 1 - E.change_ring(k).cardinality())
return v
