def curves(line):
r""" Parses one line from a curves file. Returns the label and a dict
containing fields with keys 'field_label', 'degree', 'signature',
'abs_disc', 'label', 'short_label', conductor_label',
'conductor_ideal', 'conductor_norm', 'iso_label', 'iso_nlabel', 'number',
'ainvs', 'jinv', 'cm', 'q_curve', 'base_change',
'torsion_order', 'torsion_structure', 'torsion_gens'.
Input line fields (13):
field_label conductor_label iso_label number conductor_ideal conductor_norm a1 a2 a3 a4 a6 cm base_change
Sample input line:
2.0.4.1 65.18.1 a 1 [65,18,1] 65 1,1 1,1 0,1 -1,1 -1,0 0 0
"""
# Parse the line and form the full label:
data = split(line)
if len(data) != 13:
print "line %s does not have 13 fields, skipping" % line
field_label = data[0] # string
conductor_label = data[1] # string
iso_label = data[2] # string
iso_nlabel = numerify_iso_label(iso_label) # int
number = int(data[3]) # int
short_class_label = "%s-%s" % (conductor_label, iso_label)
short_label = "%s%s" % (short_class_label, str(number))
class_label = "%s-%s" % (field_label, short_class_label)
label = "%s-%s" % (field_label, short_label)
conductor_ideal = data[4] # string
conductor_norm = int(data[5]) # int
ainvs = data[6:11] # list of 5 NFelt strings
cm = data[11] # int or '?'
if cm != '?':
cm = int(cm)
q_curve = (data[12] == '1') # bool
# Create the field and curve to compute the j-invariant:
dummy, deg, sig, abs_disc = field_data(field_label)
K = nf_lookup(field_label)
ainvsK = [parse_NFelt(K, ai) for ai in ainvs] # list of K-elements
ainvs = [[str(c) for c in ai] for ai in ainvsK]
E = EllipticCurve(ainvsK)
j = E.j_invariant()
jinv = K_list(j)
if cm == '?':
cm = get_cm(j)
if cm:
print "cm=%s for j=%s" % (cm, j)
# Here we should check that the conductor of the constructed curve
# agrees with the input conductor. We just check the norm.
if E.conductor().norm() == conductor_norm:
pass
# print "Conductor norms agree: %s" % conductor_norm
else:
raise RuntimeError("Wrong conductor for input line %s" % line)
# get torsion order, structure and generators:
# print("E = %s over %s" % (ainvsK,K))
torgroup = E.torsion_subgroup()
# print("torsion = %s" % torgroup)
ntors = int(torgroup.order())
torstruct = [int(n) for n in list(torgroup.invariants())]
torgens = [point_list(P.element()) for P in torgroup.gens()]
# get label of elliptic curve over Q for base_change cases (a
# subset of Q-curves)
if q_curve:
# print "%s is a Q-curve, testing for base-change..." % label
E1list = E.descend_to(QQ)
if len(E1list):
base_change = [cremona_to_lmfdb(E1.label()) for E1 in E1list]
print "%s is base change of %s" % (label, base_change)
else:
base_change = []
print "%s is a Q-curve, but not base-change..." % label
else:
base_change = []
return label, {
'field_label': field_label,
'degree': deg,
'signature': sig,
'abs_disc': abs_disc,
'class_label': class_label,
'short_class_label': short_class_label,
'label': label,
'short_label': short_label,
'conductor_label': conductor_label,
'conductor_ideal': conductor_ideal,
'conductor_norm': conductor_norm,
'iso_label': iso_label,
'iso_nlabel': iso_nlabel,
'number': number,
'ainvs': ainvs,
'jinv': jinv,
'cm': cm,
'q_curve': q_curve,
'base_change': base_change,
'torsion_order': ntors,
'torsion_structure': torstruct,
'torsion_gens': torgens,
}
def curves(line):
r""" Parses one line from a curves file. Returns the label and a dict
containing fields with keys 'field_label', 'degree', 'signature',
'abs_disc', 'label', 'short_label', conductor_label',
'conductor_ideal', 'conductor_norm', 'iso_label', 'number',
'ainvs', 'jinv', 'cm', 'q_curve', 'base_change',
'torsion_order', 'torsion_structure', 'torsion_gens'.
Input line fields (13):
field_label conductor_label iso_label number conductor_ideal conductor_norm a1 a2 a3 a4 a6 cm base_change
Sample input line:
2.0.4.1 [65,18,1] a 1 [65,18,1] 65 1,1 1,1 0,1 -1,1 -1,0 0 0
"""
# Parse the line and form the full label:
data = split(line)
if len(data)!=13:
print "line %s does not have 13 fields, skipping" % line
field_label = data[0] # string
conductor_label = data[1] # string
iso_label = data[2] # string
number = int(data[3]) # int
short_class_label = "%s-%s" % (conductor_label, iso_label)
short_label = "%s%s" % (short_class_label, str(number))
class_label = "%s-%s" % (field_label, short_class_label)
label = "%s-%s" % (field_label, short_label)
conductor_ideal = data[4] # string
conductor_norm = int(data[5]) # int
ainvs = data[6:11] # list of 5 NFelt strings
cm = data[11] # int or '?'
if cm!='?':
cm = int(cm)
q_curve = (data[12]=='1') # bool
# Create the field and curve to compute the j-invariant:
dummy, deg, sig, abs_disc = field_data(field_label)
K = nf_lookup(field_label)
ainvsK = [parse_NFelt(K,ai) for ai in ainvs] # list of K-elements
ainvs = [[str(c) for c in ai] for ai in ainvsK]
E = EllipticCurve(ainvsK)
j = E.j_invariant()
jinv = K_list(j)
if cm=='?':
cm = get_cm(j)
if cm:
print "cm=%s for j=%s" %(cm,j)
# Here we should check that the conductor of the constructed curve
# agrees with the input conductor. We just check the norm.
if E.conductor().norm()==conductor_norm:
pass
#print "Conductor norms agree: %s" % conductor_norm
else:
raise RuntimeError("Wrong conductor for input line %s" % line)
# get torsion order, structure and generators:
#print("E = %s over %s" % (ainvsK,K))
torgroup = E.torsion_subgroup()
#print("torsion = %s" % torgroup)
ntors = int(torgroup.order())
torstruct = [int(n) for n in list(torgroup.invariants())]
torgens = [point_list(P.element()) for P in torgroup.gens()]
# get label of elliptic curve over Q for base_change cases (a
# subset of Q-curves)
if q_curve:
#print "%s is a Q-curve, testing for base-change..." % label
E1list = E.descend_to(QQ)
if len(E1list):
base_change = [cremona_to_lmfdb(E1.label()) for E1 in E1list]
print "%s is base change of %s" % (label,base_change)
else:
base_change = []
print "%s is a Q-curve, but not base-change..." % label
else:
base_change = []
return label, {
'field_label' : field_label,
'degree': deg,
'signature': sig,
'abs_disc': abs_disc,
'class_label': class_label,
'short_class_label': short_class_label,
'label': label,
'short_label': short_label,
'conductor_label': conductor_label,
'conductor_ideal': conductor_ideal,
'conductor_norm': conductor_norm,
'iso_label': iso_label,
'number': number,
'ainvs': ainvs,
'jinv': jinv,
'cm': cm,
'q_curve': q_curve,
'base_change': base_change,
'torsion_order': ntors,
'torsion_structure': torstruct,
'torsion_gens': torgens,
}
def curves(line, verbose=False):
r""" Parses one line from a curves file. Returns the label and a dict
containing fields with keys 'field_label', 'degree', 'signature',
'abs_disc', 'label', 'short_label', conductor_label',
'conductor_ideal', 'conductor_norm', 'iso_label', 'iso_nlabel',
'number', 'ainvs', 'jinv', 'cm', 'q_curve', 'base_change',
'torsion_order', 'torsion_structure', 'torsion_gens'; and (added
May 2016): 'equation', 'local_data', 'non_min_p', 'minD'
Input line fields (13):
field_label conductor_label iso_label number conductor_ideal conductor_norm a1 a2 a3 a4 a6 cm base_change
Sample input line:
2.0.4.1 65.18.1 a 1 [65,18,1] 65 1,1 1,1 0,1 -1,1 -1,0 0 0
"""
# Parse the line and form the full label:
data = split(line)
if len(data) != 13:
print "line %s does not have 13 fields, skipping" % line
field_label = data[0] # string
IQF_flag = field_label.split(".")[:2] == ['2', '0']
K = nf_lookup(field_label) if IQF_flag else None
conductor_label = data[1] # string
# convert label (does nothing except for imaginary quadratic)
conductor_label = convert_conductor_label(field_label, conductor_label)
iso_label = data[2] # string
iso_nlabel = numerify_iso_label(iso_label) # int
number = int(data[3]) # int
short_class_label = "%s-%s" % (conductor_label, iso_label)
short_label = "%s%s" % (short_class_label, str(number))
class_label = "%s-%s" % (field_label, short_class_label)
label = "%s-%s" % (field_label, short_label)
conductor_ideal = data[4] # string
conductor_norm = int(data[5]) # int
ainvs = ";".join(data[6:11]) # one string joining 5 NFelt strings
cm = data[11] # int or '?'
if cm != '?':
cm = int(cm)
# Create the field and curve to compute the j-invariant:
dummy, deg, sig, abs_disc = field_data(field_label)
K = nf_lookup(field_label)
#print("Field %s created, gen_name = %s" % (field_label,str(K.gen())))
ainvsK = parse_ainvs(K, ainvs) # list of K-elements
E = EllipticCurve(ainvsK)
#print("{} created with disc = {}, N(disc)={}".format(E,K.ideal(E.discriminant()).factor(),E.discriminant().norm().factor()))
j = E.j_invariant()
jinv = NFelt(j)
if cm == '?':
cm = get_cm(j)
if cm:
print "cm=%s for j=%s" % (cm, j)
q_curve = data[12] # 0, 1 or ?. If unknown we'll determine this below.
if q_curve in ['0', '1']: # already set -- easy
q_curve = bool(int(q_curve))
else:
try:
q_curve = is_Q_curve(E)
except NotImplementedError:
q_curve = '?'
# Here we should check that the conductor of the constructed curve
# agrees with the input conductor.
N = ideal_from_string(K, conductor_ideal)
NE = E.conductor()
if N == "wrong" or N != NE:
print(
"Wrong conductor ideal {} for label {}, using actual conductor {} instead"
.format(conductor_ideal, label, NE))
conductor_ideal = ideal_to_string(NE)
N = NE
# get torsion order, structure and generators:
torgroup = E.torsion_subgroup()
ntors = int(torgroup.order())
torstruct = [int(n) for n in list(torgroup.invariants())]
torgens = [point_string(P.element()) for P in torgroup.gens()]
# get label of elliptic curve over Q for base_change cases (a
# subset of Q-curves)
if True: # q_curve: now we have not precomputed Q-curve status
# but still want to test for base change!
if verbose:
print("testing {} for base-change...".format(label))
E1list = E.descend_to(QQ)
if len(E1list):
base_change = [cremona_to_lmfdb(E1.label()) for E1 in E1list]
if verbose:
print "%s is base change of %s" % (label, base_change)
else:
base_change = []
# print "%s is a Q-curve, but not base-change..." % label
else:
base_change = []
# NB if this is not a global minimal model then local_data may
# include a prime at which we have good reduction. This causes no
# problems except that the bad_reduction_type is then None which
# cannot be converted to an integer. The bad reduction types are
# coded as (Sage) integers in {-1,0,1}.
local_data = [{
'p':
ideal_to_string(ld.prime()),
'normp':
str(ld.prime().norm()),
'ord_cond':
int(ld.conductor_valuation()),
'ord_disc':
int(ld.discriminant_valuation()),
'ord_den_j':
int(max(0, -(E.j_invariant().valuation(ld.prime())))),
'red':
None
if ld.bad_reduction_type() == None else int(ld.bad_reduction_type()),
'kod':
web_latex(ld.kodaira_symbol()).replace('$', ''),
'cp':
int(ld.tamagawa_number())
} for ld in E.local_data()]
non_minimal_primes = [ideal_to_string(P) for P in E.non_minimal_primes()]
minD = ideal_to_string(E.minimal_discriminant_ideal())
edata = {
'field_label': field_label,
'degree': deg,
'signature': sig,
'abs_disc': abs_disc,
'class_label': class_label,
'short_class_label': short_class_label,
'label': label,
'short_label': short_label,
'conductor_label': conductor_label,
'conductor_ideal': conductor_ideal,
'conductor_norm': conductor_norm,
'iso_label': iso_label,
'iso_nlabel': iso_nlabel,
'number': number,
'ainvs': ainvs,
'jinv': jinv,
'cm': cm,
'q_curve': q_curve,
'base_change': base_change,
'torsion_order': ntors,
'torsion_structure': torstruct,
'torsion_gens': torgens,
'equation': web_latex(E),
'local_data': local_data,
'minD': minD,
'non_min_p': non_minimal_primes,
}
return label, edata
def curves(line, verbose=False):
r""" Parses one line from a curves file. Returns the label and a dict
containing fields with keys 'field_label', 'degree', 'signature',
'abs_disc', 'label', 'short_label', conductor_label',
'conductor_ideal', 'conductor_norm', 'iso_label', 'iso_nlabel',
'number', 'ainvs', 'jinv', 'cm', 'q_curve', 'base_change',
'torsion_order', 'torsion_structure', 'torsion_gens'; and (added
May 2016): 'equation', 'local_data', 'non_min_p', 'minD'
Input line fields (13):
field_label conductor_label iso_label number conductor_ideal conductor_norm a1 a2 a3 a4 a6 cm base_change
Sample input line:
2.0.4.1 65.18.1 a 1 [65,18,1] 65 1,1 1,1 0,1 -1,1 -1,0 0 0
"""
# Parse the line and form the full label:
data = split(line)
if len(data) != 13:
print "line %s does not have 13 fields, skipping" % line
field_label = data[0] # string
IQF_flag = field_label.split(".")[:2] == ['2','0']
K = nf_lookup(field_label) if IQF_flag else None
conductor_label = data[1] # string
# convert label (does nothing except for imaginary quadratic)
conductor_label = convert_conductor_label(field_label, conductor_label)
iso_label = data[2] # string
iso_nlabel = numerify_iso_label(iso_label) # int
number = int(data[3]) # int
short_class_label = "%s-%s" % (conductor_label, iso_label)
short_label = "%s%s" % (short_class_label, str(number))
class_label = "%s-%s" % (field_label, short_class_label)
label = "%s-%s" % (field_label, short_label)
conductor_ideal = data[4] # string
conductor_norm = int(data[5]) # int
ainvs = ";".join(data[6:11]) # one string joining 5 NFelt strings
cm = data[11] # int or '?'
if cm != '?':
cm = int(cm)
q_curve = (data[12] == '1') # bool
# Create the field and curve to compute the j-invariant:
dummy, deg, sig, abs_disc = field_data(field_label)
K = nf_lookup(field_label)
#print("Field %s created, gen_name = %s" % (field_label,str(K.gen())))
ainvsK = parse_ainvs(K,ainvs) # list of K-elements
E = EllipticCurve(ainvsK)
#print("{} created with disc = {}, N(disc)={}".format(E,K.ideal(E.discriminant()).factor(),E.discriminant().norm().factor()))
j = E.j_invariant()
jinv = NFelt(j)
if cm == '?':
cm = get_cm(j)
if cm:
print "cm=%s for j=%s" % (cm, j)
# Here we should check that the conductor of the constructed curve
# agrees with the input conductor.
N = ideal_from_string(K,conductor_ideal)
NE = E.conductor()
if N=="wrong" or N!=NE:
print("Wrong conductor ideal {} for label {}, using actual conductor {} instead".format(conductor_ideal,label,NE))
conductor_ideal = ideal_to_string(NE)
N = NE
# get torsion order, structure and generators:
torgroup = E.torsion_subgroup()
ntors = int(torgroup.order())
torstruct = [int(n) for n in list(torgroup.invariants())]
torgens = [point_string(P.element()) for P in torgroup.gens()]
# get label of elliptic curve over Q for base_change cases (a
# subset of Q-curves)
if True: # q_curve: now we have not precomputed Q-curve status
# but still want to test for base change!
if verbose:
print("testing {} for base-change...".format(label))
E1list = E.descend_to(QQ)
if len(E1list):
base_change = [cremona_to_lmfdb(E1.label()) for E1 in E1list]
if verbose:
print "%s is base change of %s" % (label, base_change)
else:
base_change = []
# print "%s is a Q-curve, but not base-change..." % label
else:
base_change = []
# NB if this is not a global minimal model then local_data may
# include a prime at which we have good reduction. This causes no
# problems except that the bad_reduction_type is then None which
# cannot be converted to an integer. The bad reduction types are
# coded as (Sage) integers in {-1,0,1}.
local_data = [{'p': ideal_to_string(ld.prime()),
'normp': str(ld.prime().norm()),
'ord_cond':int(ld.conductor_valuation()),
'ord_disc':int(ld.discriminant_valuation()),
'ord_den_j':int(max(0,-(E.j_invariant().valuation(ld.prime())))),
'red':None if ld.bad_reduction_type()==None else int(ld.bad_reduction_type()),
'kod':web_latex(ld.kodaira_symbol()).replace('$',''),
'cp':int(ld.tamagawa_number())}
for ld in E.local_data()]
non_minimal_primes = [ideal_to_string(P) for P in E.non_minimal_primes()]
minD = ideal_to_string(E.minimal_discriminant_ideal())
edata = {
'field_label': field_label,
'degree': deg,
'signature': sig,
'abs_disc': abs_disc,
'class_label': class_label,
'short_class_label': short_class_label,
'label': label,
'short_label': short_label,
'conductor_label': conductor_label,
'conductor_ideal': conductor_ideal,
'conductor_norm': conductor_norm,
'iso_label': iso_label,
'iso_nlabel': iso_nlabel,
'number': number,
'ainvs': ainvs,
'jinv': jinv,
'cm': cm,
'q_curve': q_curve,
'base_change': base_change,
'torsion_order': ntors,
'torsion_structure': torstruct,
'torsion_gens': torgens,
'equation': web_latex(E),
'local_data': local_data,
'minD': minD,
'non_min_p': non_minimal_primes,
}
return label, edata