def endo_statement_isog(factorsQQ, factorsRR, fieldstring):
statement = """<table class="g2">"""
factorsQQ_number = len(factorsQQ)
factorsQQ_pretty = [ field_pretty(fac[0]) for fac in factorsQQ if
fac[0] ]
# First row: description of endomorphism algebra factors
statement += """<tr><td>\(\End (J_{%s}) \otimes \Q \)</td><td>\(\simeq\)</td><td>"""\
% fieldstring
# In the case of only one factor we either get a number field or a
# quaternion algebra:
if factorsQQ_number == 1:
# First we deal with the number field case,
# in which we have set the discriminant to be -1
if factorsQQ[0][2] == -1:
# Prettify if labels available, otherwise return defining polynomial:
if factorsQQ_pretty:
statement += """<a href=%s>%s</a>"""\
% (url_for("number_fields.by_label",
label=factorsQQ[0][0]), factorsQQ_pretty[0])
else:
statement += """number field with defining polynomial \(%s\)"""\
% intlist_to_poly(factorsQQ[0][1])
# Detect CM by presence of a quartic polynomial:
# TODO: Add knowl link
if len(factorsQQ[0][1]) == 5:
statement += """(CM)"""
# Up next is the case of a matrix ring (trivial disciminant), with
# labels and full prettification always available:
elif factorsQQ[0][2] == 1:
statement += """\(\mathrm{M}_2(\)<a href=%s>%s</a>\()\)"""\
% (url_for("number_fields.by_label", label=factorsQQ[0][0]),
factorsQQ_pretty[0])
# And finally we deal with quaternion algebras over the rationals:
else:
statement += """quaternion algebra over <a href=%s>%s</a> of discriminant %s"""\
% (url_for("number_fields.by_label", label=factorsQQ[0][0]),
factorsQQ_pretty[0], factorsQQ[0][2])
# If there are two factors, then we get two at most quadratic fields:
else:
statement += """<a href=%s>%s</a> \(\\times\) <a href=%s>%s</a>"""\
% (url_for("number_fields.by_label", label=factorsQQ[0][0]),
factorsQQ_pretty[0], url_for("number_fields.by_label",
label=factorsQQ[1][0]), factorsQQ_pretty[1])
# End of first row:
statement += """</td></tr>"""
# Second row: description of algebra tensored with RR
statement += """<tr><td>\(\End (J_{%s}) \otimes \R\)</td><td>\(\simeq\)</td> <td>\(%s\)</td></tr>"""\
% (fieldstring, factorsRR_raw_to_pretty(factorsRR))
# End of statement:
statement += """</table>"""
return statement
def make_class(self):
from lmfdb.genus2_curves.genus2_curve import url_for_curve_label
# Data
curves_data = g2cdb().curves.find({"class" : self.label},{'_id':int(0),'label':int(1),'min_eqn':int(1),'disc_key':int(1)}).sort([("disc_key", ASCENDING), ("label", ASCENDING)])
assert curves_data
self.curves = [ {"label" : c['label'], "equation_formatted" : list_to_min_eqn(c['min_eqn']),
"url": url_for_curve_label(c['label'])} for c in curves_data ]
self.ncurves = curves_data.count()
self.bad_lfactors = [ [c[0], list_to_factored_poly_otherorder(c[1])]
for c in self.bad_lfactors]
# Data derived from Sato-Tate group
self.st_group_name = st_group_name(self.st_group)
self.st_group_href = st_group_href(self.st_group)
self.st0_group_name = st0_group_name(self.real_geom_end_alg)
# Later used in Lady Gaga box:
self.real_geom_end_alg_disp = [r'\End(J_{\overline{\Q}}) \otimes \R',
end_alg_name(self.real_geom_end_alg)]
if self.is_gl2_type:
self.is_gl2_type_name = 'yes'
else:
self.is_gl2_type_name = 'no'
# Endomorphism data
endodata = g2cdb().endomorphisms.find_one({"label" :
self.curves[0]['label']})
self.gl2_statement_base = \
gl2_statement_base(endodata['factorsRR_base'], r'\(\Q\)')
self.endo_statement_base = \
"""Endomorphism algebra over \(\Q\):<br>""" + \
endo_statement_isog(endodata['factorsQQ_base'],
endodata['factorsRR_base'], r'')
endodata['fod_poly'] = intlist_to_poly(endodata['fod_coeffs'])
self.fod_statement = fod_statement(endodata['fod_label'],
endodata['fod_poly'])
if endodata['fod_label'] != '1.1.1.1':
self.endo_statement_geom = \
"""Endomorphism algebra over \(\overline{\Q}\):<br>""" + \
endo_statement_isog(endodata['factorsQQ_geom'],
endodata['factorsRR_geom'], r'\overline{\Q}')
else:
self.endo_statement_geom = ''
# Title
self.title = "Genus 2 Isogeny Class %s" % (self.label)
# Lady Gaga box
self.properties = (
('Label', self.label),
('Number of curves', str(self.ncurves)),
('Conductor','%s' % self.cond),
('Sato-Tate group', self.st_group_href),
('\(%s\)' % self.real_geom_end_alg_disp[0],
'\(%s\)' % self.real_geom_end_alg_disp[1]),
('\(\mathrm{GL}_2\)-type','%s' % self.is_gl2_type_name)
)
x = self.label.split('.')[1]
self.friends = [('L-function', url_for("l_functions.l_function_genus2_page", cond=self.cond,x=x))]
#self.downloads = [('Download Euler factors', ".")]
#self.downloads = [
# ('Download Euler factors', "."),
# url_for(".download_g2c_eulerfactors", label=self.label)),
# ('Download stored data for all curves',
# url_for(".download_g2c_all", label=self.label))
# ]
# Breadcrumbs
self.bread = (
('Genus 2 Curves', url_for(".index")),
('$\Q$', url_for(".index_Q")),
('%s' % self.cond, url_for(".by_conductor", cond=self.cond)),
('%s' % self.label, ' ')
)
# More friends (NOTE: to be improved)
self.ecproduct_wurl = []
if hasattr(self, 'ecproduct'):
for i in range(2):
curve_label = self.ecproduct[i]
crv_url = url_for("ec.by_ec_label", label=curve_label)
if i == 1 or len(set(self.ecproduct)) != 1:
self.friends.append(('Elliptic curve ' + curve_label,
crv_url))
self.ecproduct_wurl.append({'label' : curve_label, 'url' :
crv_url})
self.ecquadratic_wurl = []
if hasattr(self, 'ecquadratic'):
for i in range(len(self.ecquadratic)):
curve_label = self.ecquadratic[i]
crv_spl = curve_label.split('-')
crv_url = url_for("ecnf.show_ecnf_isoclass", nf = crv_spl[0],
conductor_label = crv_spl[1], class_label = crv_spl[2])
self.friends.append(('Elliptic curve ' + curve_label, crv_url))
self.ecquadratic_wurl.append({'label' : curve_label, 'url' :
crv_url, 'nf' : crv_spl[0]})
if hasattr(self, 'mfproduct'):
for i in range(len(self.mfproduct)):
mf_label = self.mfproduct[i]
mf_spl = mf_label.split('.')
mf_spl.append(mf_spl[2][-1])
mf_spl[2] = mf_spl[2][:-1] # Need a splitting function
mf_url = url_for("emf.render_elliptic_modular_forms",
level=mf_spl[0], weight=mf_spl[1], character=mf_spl[2],
label=mf_spl[3])
self.friends.append(('Modular form ' + mf_label, mf_url))
if hasattr(self, 'mfhilbert'):
for i in range(len(self.mfhilbert)):
mf_label = self.mfhilbert[i]
mf_spl = mf_label.split('-')
mf_url = url_for("hmf.render_hmf_webpage",
field_label=mf_spl[0], label=mf_label)
self.friends.append(('Hilbert modular form ' + mf_label, mf_url))
