def MWShaInfo(E, HeightBound=None, test_saturation=False, verbose=False):
r"""
Interface to Magma's MordellWeilShaInformation function
INPUT:
- E : an elliptic curve defined over a number field (including Q)
OUTPUT:
a triple [rank_low_up, gens, sha_bound_dict] where
- rank_bounds is a list of 2 integers [r1,r2] such r1 <= rank(E) <= r2
- gens is a list of r1 independent points on E
- sha_bound_dict is a dict with keys positive integers n, values rank bounds for Sha[n].
EXAMPLE::
sage: E = EllipticCurve('5077a1')
sage: MWShaInfo(E)
[[3, 3], [(2 : -1 : 1), (9/4 : -15/8 : 1), (-1 : 3 : 1)], {2: [0, 0]}]
"""
K = E.base_field()
def convert_point(P):
return E([K(c.sage()) for c in P.Eltseq()])
if verbose:
print("calling magma...")
from sage.all import magma
if HeightBound is None:
MWSI = magma(E).MordellWeilShaInformation(nvals=3)
else:
MWSI = magma(E).MordellWeilShaInformation(nvals=3,
HeightBound=HeightBound)
if verbose:
print("...done.")
rank_bounds = MWSI[0].sage()
gens = [convert_point(P) for P in MWSI[1]]
sha_bound_dict = dict(MWSI[2].sage())
if gens and test_saturation:
if verbose:
print("testing that Magma's generators are saturated...")
newgens, index, newreg = E.saturation(gens, verbose)
if index > 1:
# Must print this even if not verbose!
print(
"Magma's generators for curve %s were not saturated! index = %s"
% (E.ainvs(), index))
gens = newgens
else:
if verbose:
print("... and they are!")
return [rank_bounds, gens, sha_bound_dict]
def MWShaInfo(E, HeightBound=None, test_saturation=False, verbose=False):
r"""
Interface to Magma's MordellWeilShaInformation function
INPUT:
- E : an elliptic curve defined over a number field (including Q)
OUTPUT:
a triple [rank_low_up, gens, sha_bound_dict] where
- rank_bounds is a list of 2 integers [r1,r2] such r1 <= rank(E) <= r2
- gens is a list of r1 independent points on E
- sha_bound_dict is a dict with keys positive integers n, values rank bounds for Sha[n].
EXAMPLE::
sage: E = EllipticCurve('5077a1')
sage: MWShaInfo(E)
[[3, 3], [(2 : -1 : 1), (9/4 : -15/8 : 1), (-1 : 3 : 1)], {2: [0, 0]}]
"""
K = E.base_field()
def convert_point(P):
return E([K(c.sage()) for c in P.Eltseq()])
if verbose:
print("calling magma...")
from sage.all import magma
if HeightBound is None:
MWSI = magma(E).MordellWeilShaInformation(nvals=3)
else:
MWSI = magma(E).MordellWeilShaInformation(nvals=3, HeightBound=HeightBound)
if verbose:
print("...done.")
rank_bounds = MWSI[0].sage()
gens = [convert_point(P) for P in MWSI[1]]
sha_bound_dict = dict(MWSI[2].sage())
if gens and test_saturation:
if verbose:
print("testing that Magma's generators are saturated...")
newgens, index, newreg = E.saturation(gens, verbose)
if index > 1:
# Must print this even if not verbose!
print("Magma's generators for curve %s were not saturated! index = %s" % (E.ainvs(), index))
gens = newgens
else:
if verbose:
print("... and they are!")
return [rank_bounds, gens, sha_bound_dict]
def __init__(self, X, prec, bound=0, molin_neurohr=False, periods=""):
self.X = X
self.g = magma.Genus(self.X)
self.F = magma.BaseRing(self.X)
self.prec = magma(prec)
self.bound = magma(bound)
self.molin_neurohr = magma(molin_neurohr)
self._eqsCC_ = magma.EmbedCurveEquations(self.X, self.prec)
self._eqsF_ = magma.DefiningEquations(self.X)
if periods:
self._P_ = periods
else:
self._P_ = self.period_matrix()
self._calculate_geometric_representation_()
def optimize_representations(self):
self._calculate_dictionary_()
for dict_pair in self._dict_['entries']:
structure = dict_pair['structure']
rep = magma(
[gen['tangent'] for gen in structure['representation']])
optrep = Optimize_Representation(rep)
for i in range(len(optrep)):
structure['representation'][i]['tangent'] = optrep[i + 1]
def representations(self):
self._calculate_dictionary_()
list_to_fill = []
for dict_pair in self._dict_['entries']:
dict_to_fill = dict()
dict_to_fill['field'] = dict_pair['field']['magma']
dict_to_fill['representation'] = magma([
gen['tangent']
for gen in dict_pair['structure']['representation']
])
list_to_fill.append(dict_to_fill)
return list_to_fill
def __init__(self, Endo, K="geometric"):
self.X = Endo.X
self.g = Endo.g
self.F = Endo.F
self._P_ = Endo._P_
self._geo_rep_list_ = Endo._geo_rep_list_
if K == "geometric":
self.field = Endo.endomorphism_field()
elif K == "base":
self.field = self.F
else:
self.field = magma(K)
self._list_ = magma.EndomorphismStructure(self._geo_rep_list_,
self.field, self.F)
self._desc_ = desc_structure(self._list_)
self._index_dict_ = index_dictionary()
def mPlaneCurve(f):
return magma.PlaneCurve(magma(f))
def mHyperellipticCurve(f, h):
return magma.HyperellipticCurve(magma(f), magma(h))
def representation(self):
self._calculate_dictionary_()
return magma([gen['tangent'] for gen in self._dict_['representation']])
def mPlaneCurve(f, prec):
return magma.PlaneCurveExtra(magma(f), magma(prec))
def mHyperellipticCurve(f, h, prec):
return magma.HyperellipticCurveExtra(magma(f), magma(h), magma(prec))