def projective_field(self):
projfield = self._data['Proj_Polynomial']
if projfield == [0]:
return 'data not computed'
if projfield == [0, 1]:
return formatfield(projfield)
return 'Galois closure of ' + formatfield(projfield)
def projective_field(self):
projfield = self._data['Proj_Polynomial']
if projfield == [0]:
return 'data not computed'
if projfield == [0,1]:
return formatfield(projfield)
return formatfield(projfield, missing_text="Degree %s field"%(len(projfield)-1))
def kerfield(self):
kerpoly = self.kernel_field_poly
if kerpoly and self.order <= 100:
return formatfield(
kerpoly,
missing_text="Number field defined by a degree %d polynomial" %
self.order)
else:
return "Number field defined by a degree %d polynomial (not computed)" % self.order
def make_torsion_growth(self):
# The torsion growth table has one row per extension field
tgdata = list(db.ec_torsion_growth.search({'lmfdb_label': self.lmfdb_label}))
if not tgdata: # we only have torsion growth data for some range of conductors
self.torsion_growth_data_exists = False
return
self.torsion_growth_data_exists = True
self.tg = tg = {}
tg['data'] = tgextra = []
# find all base changes of this curve in the database, if any:
bcs = list(db.ec_nfcurves.search({'base_change': {'$contains': [self.lmfdb_label]}}, projection='label'))
# extract the fields from the labels of the base-change curves:
bc_fields = [lab.split("-")[0] for lab in bcs]
bc_pols = [db.nf_fields.lookup(lab, projection='coeffs') for lab in bc_fields]
tg['fields_missing'] = False
for tgd in tgdata:
tg1 = {}
tg1['bc_label'] = "Not in database"
tg1['d'] = tgd['degree']
F = tgd['field']
tg1['f'] = formatfield(F)
if "missing" in tg1['f']:
tg['fields_missing'] = True
T = tgd['torsion']
tg1['t'] = r'\(' + r' \times '.join(r'\Z/{}\Z'.format(n) for n in T) + r'\)'
bcc = next((lab for lab, pol in zip(bcs, bc_pols) if pol==F), None)
if bcc:
from lmfdb.ecnf.main import split_full_label
F, NN, I, C = split_full_label(bcc)
tg1['bc_label'] = bcc
tg1['bc_url'] = url_for('ecnf.show_ecnf', nf=F, conductor_label=NN, class_label=I, number=C)
tg1['m'] = 0 # holds multiplicity per degree
tgextra.append(tg1)
tgextra.sort(key = lambda x: x['d'])
tg['n'] = len(tgextra)
lastd = 1
for tg1 in tgextra:
d = tg1['d']
if d!=lastd:
tg1['m'] = len([x for x in tgextra if x['d']==d])
lastd = d
## Hard-coded this for now. Note that the *only* place where
## this number is used is in the ec-curve template where it
## says "The number fields ... of degree less than
## {{data.tg.maxd}} such that...".
tg['maxd'] = 24
def field_knowl(self):
nfgg = self.number_field_galois_group()
return formatfield(nfgg.polynomial())
