def make_object(self, curve, endo, tama, ratpts, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = str(curve['Lhash'])
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex(factor(int(
data['cond']))).replace(r"-1 \cdot", "-")
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['mw_rank'] = ZZ(0) if curve.get('mw_rank') is None else ZZ(
curve['mw_rank']) # 0 will be marked as a lower bound
data['mw_rank_proved'] = curve['mw_rank_proved']
data['analytic_rank_proved'] = curve['analytic_rank_proved']
data['hasse_weil_proved'] = curve['hasse_weil_proved']
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1, 4, data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page",
cond=data['slabel'][0],
x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [(c[0],
list_to_factored_poly_otherorder(c[1]))
for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['abs_disc'])
data['disc'] = curve['disc_sign'] * data['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = min_eqns_pretty(data['min_eqn'])
data['disc_factor_latex'] = web_latex(factor(
data['disc'])).replace(r"-1 \cdot", "-")
data['igusa_clebsch'] = [
ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])
]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [
web_latex(zfactor(i)).replace(r"-1 \cdot", "-")
for i in data['igusa_clebsch']
]
data['igusa_factor_latex'] = [
web_latex(zfactor(j)).replace(r"-1 \cdot", "-")
for j in data['igusa']
]
data['aut_grp'] = small_group_label_display_knowl(
'%d.%d' % tuple(literal_eval(curve['aut_grp_id'])))
data['geom_aut_grp'] = small_group_label_display_knowl(
'%d.%d' % tuple(literal_eval(curve['geom_aut_grp_id'])))
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['has_square_sha'] = "square" if curve[
'has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",", 1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['torsion_order'] = curve['torsion_order']
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['real_period'] = decimal_pretty(str(curve['real_period']))
data['regulator'] = decimal_pretty(
str(curve['regulator']
)) if curve['regulator'] > -0.5 else 'unknown'
if data['mw_rank'] == 0 and data['mw_rank_proved']:
data['regulator'] = '1' # display an exact 1 when we know this
data['tamagawa_product'] = ZZ(
curve['tamagawa_product']) if curve.get(
'tamagawa_product') else 0
data['analytic_sha'] = ZZ(
curve['analytic_sha']) if curve.get('analytic_sha') else 0
data['leading_coeff'] = decimal_pretty(
str(curve['leading_coeff']
)) if curve['leading_coeff'] else 'unknown'
data['rat_pts'] = ratpts['rat_pts']
data['rat_pts_v'] = ratpts['rat_pts_v']
data['rat_pts_table'] = ratpts_table(ratpts['rat_pts'],
ratpts['rat_pts_v'])
data['mw_gens_v'] = ratpts['mw_gens_v']
lower = len([n for n in ratpts['mw_invs'] if n == 0])
upper = data['analytic_rank']
invs = ratpts[
'mw_invs'] if data['mw_gens_v'] or lower >= upper else [
0 for n in range(upper - lower)
] + ratpts['mw_invs']
if len(invs) == 0:
data['mw_group'] = 'trivial'
else:
data['mw_group'] = r'\(' + r' \times '.join([
(r'\Z' if n == 0 else r'\Z/{%s}\Z' % n) for n in invs
]) + r'\)'
if lower >= upper:
data['mw_gens_table'] = mw_gens_table(ratpts['mw_invs'],
ratpts['mw_gens'],
ratpts['mw_heights'],
ratpts['rat_pts'])
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl(
curve['two_torsion_field'][0],
field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data[
'two_torsion_field_knowl'] = r"splitting field of \(%s\) with Galois group %s" % (
intlist_to_poly(
t[1]), group_display_knowl(t[2][0], t[2][1]))
tamalist = [[item['p'], item['tamagawa_number']] for item in tama]
data['local_table'] = local_table(data['abs_disc'], data['cond'],
tamalist,
data['bad_lfactors_pretty'])
else:
# invariants specific to isogeny class
curves_data = list(
db.g2c_curves.search({"class": curve['class']},
['label', 'eqn']))
if not curves_data:
raise KeyError(
"No curves found in database for isogeny class %s of genus 2 curve %s."
% (curve['class'], curve['label']))
data['curves'] = [{
"label":
c['label'],
"equation_formatted":
min_eqn_pretty(literal_eval(c['eqn'])),
"url":
url_for_curve_label(c['label'])
} for c in curves_data]
lfunc_data = db.lfunc_lfunctions.lucky(
{'Lhash': str(curve['Lhash'])})
if not lfunc_data:
raise KeyError(
"No Lfunction found in database for isogeny class of genus 2 curve %s."
% curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [
[nth_prime(n + 1), lfunc_data['euler_factors'][n]]
for n in range(len(lfunc_data['euler_factors']))
if nth_prime(n + 1) < 30 and (data['cond'] %
nth_prime(n + 1))
]
data['good_lfactors_pretty'] = [
(c[0], list_to_factored_poly_otherorder(c[1]))
for c in data['good_lfactors']
]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(
endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = (
r"Endomorphism %s over \(\Q\):<br>" %
("ring" if is_curve else "algebra") +
end_statement(data['factorsQQ_base'],
endo['factorsRR_base'],
ring=data['end_ring_base'] if is_curve else None))
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(
data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(
data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = (
r"Endomorphism %s over \(\overline{\Q}\):" %
("ring" if is_curve else "algebra") + end_statement(
data['factorsQQ_geom'],
data['factorsRR_geom'],
field=r'\overline{\Q}',
ring=data['end_ring_geom'] if is_curve else None))
data['real_geom_end_alg_name'] = real_geom_end_alg_name(
curve['real_geom_end_alg'])
data['geom_end_alg_name'] = geom_end_alg_name(curve['geom_end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(
data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(
data['is_simple_geom'], data['split_field_label'],
data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(
endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'],
data.get('split_labels'),
data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
plot_from_db = db.g2c_plots.lucky({"label": curve['label']})
if (plot_from_db is None):
self.plot = encode_plot(
eqn_list_to_curve_plot(
data['min_eqn'], ratpts['rat_pts'] if ratpts else []))
else:
self.plot = plot_from_db['plot']
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(
self.plot)
properties += [
(None, plot_link),
('Conductor', str(data['cond'])),
('Discriminant', str(data['disc'])),
]
if data['mw_rank_proved']:
properties += [('Mordell-Weil group', data['mw_group'])]
properties += [
('Sato-Tate group', data['st_group_link']),
(r'\(\End(J_{\overline{\Q}}) \otimes \R\)',
r'\(%s\)' % data['real_geom_end_alg_name']),
(r'\(\End(J_{\overline{\Q}}) \otimes \Q\)',
r'\(%s\)' % data['geom_end_alg_name']),
(r'\(\overline{\Q}\)-simple', bool_pretty(data['is_simple_geom'])),
(r'\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = []
if is_curve:
friends.append(('Isogeny class %s.%s' %
(data['slabel'][0], data['slabel'][1]),
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1])))
# first deal with EC
ecs = []
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
ecs.append(("Elliptic curve " + friend_label,
url_for_ec(friend_label)))
else:
ecs.append(
("Isogeny class " + ec_label_class(friend_label),
url_for_ec_class(friend_label)))
ecs.sort(key=lambda x: key_for_numerically_sort(x[0]))
# then again EC from lfun
instances = []
for elt in db.lfunc_instances.search(
{
'Lhash': data['Lhash'],
'type': 'ECQP'
}, 'url'):
instances.extend(elt.split('|'))
# and then the other isogeny friends
instances.extend([
elt['url'] for elt in get_instances_by_Lhash_and_trace_hash(
data["Lhash"], 4, int(data["Lhash"]))
])
exclude = {
elt[1].rstrip('/').lstrip('/')
for elt in self.friends if elt[1]
}
exclude.add(data['lfunc_url'].lstrip('/L/').rstrip('/'))
for elt in ecs + names_and_urls(instances, exclude=exclude):
# because of the splitting we must use G2C specific code
add_friend(friends, elt)
if is_curve:
friends.append(('Twists',
url_for(".index_Q",
g20=str(data['g2'][0]),
g21=str(data['g2'][1]),
g22=str(data['g2'][2]))))
friends.append(('L-function', data['lfunc_url']))
# Breadcrumbs
self.bread = bread = [('Genus 2 Curves', url_for(".index")),
(r'$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0],
url_for(".by_conductor",
cond=data['slabel'][0])),
('%s' % data['slabel'][1],
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1]))]
if is_curve:
bread += [('%s' % data['slabel'][2],
url_for(".by_url_isogeny_class_discriminant",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2])),
('%s' % data['slabel'][3],
url_for(".by_url_curve_label",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2],
num=data['slabel'][3]))]
# Title
self.title = "Genus 2 " + ("Curve " if is_curve else
"Isogeny Class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage': '', 'magma': ''} # use default show names
f, h = fh = data['min_eqn']
g = simplify_hyperelliptic(fh)
code['curve'] = {
'sage':
'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s));'
% (f, h),
'magma':
'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'
% (f, h)
}
code['simple_curve'] = {
'sage': 'X = HyperellipticCurve(R(%s))' % (g),
'magma': 'X,pi:= SimplifiedModel(C);'
}
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation(' + str(
data['cond']) + ',2),R!' + str(bad2) + '>*]'
else:
magma_cond_option = ''
code['cond'] = {
'magma':
'Conductor(LSeries(C%s)); Factorization($1);' % magma_cond_option
}
code['disc'] = {
'magma': 'Discriminant(C); Factorization(Integers()!$1);'
}
code['geom_inv'] = {
'sage':
'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':
'IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);'
}
code['aut'] = {'magma': 'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {
'magma':
'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'
}
code['num_rat_wpts'] = {
'magma': '#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'
}
if ratpts:
code['rat_pts'] = {
'magma':
'[' + ','.join([
"C![%s,%s,%s]" % (p[0], p[1], p[2])
for p in ratpts['rat_pts']
]) + '];'
}
code['mw_group'] = {'magma': 'MordellWeilGroupGenus2(Jacobian(C));'}
code['two_selmer'] = {
'magma': 'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'
}
code['has_square_sha'] = {'magma': 'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {
'magma':
'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'
}
code['torsion_subgroup'] = {
'magma':
'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'
}
def make_object(self, curve, endo, tama, ratpts, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = str(curve['Lhash'])
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex(factor(int(data['cond'])))
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1,4,data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page", cond=data['slabel'][0], x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['abs_disc'])
data['disc'] = curve['disc_sign'] * data['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = list_to_min_eqn(data['min_eqn'])
data['disc_factor_latex'] = web_latex(factor(data['disc']))
data['igusa_clebsch'] = [ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [web_latex(zfactor(i)) for i in data['igusa_clebsch']]
data['igusa_factor_latex'] = [ web_latex(zfactor(j)) for j in data['igusa'] ]
data['aut_grp_id'] = curve['aut_grp_id']
data['geom_aut_grp_id'] = curve['geom_aut_grp_id']
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['has_square_sha'] = "square" if curve['has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",",1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['torsion_order'] = curve['torsion_order']
data['torsion_factors'] = [ZZ(a) for a in literal_eval(curve['torsion_subgroup'])]
if len(data['torsion_factors']) == 0:
data['torsion_subgroup'] = '\mathrm{trivial}'
else:
data['torsion_subgroup'] = ' \\times '.join([ '\Z/{%s}\Z' % n for n in data['torsion_factors'] ])
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['tama'] = ''
for item in tama:
if item['tamagawa_number'] > 0:
tamgwnr = str(item['tamagawa_number'])
else:
tamgwnr = 'N/A'
data['tama'] += tamgwnr + ' (p = ' + str(item['p']) + '), '
data['tama'] = data['tama'][:-2] # trim last ", "
if ratpts:
if len(ratpts['rat_pts']):
data['rat_pts'] = ', '.join(web_latex('(' +' : '.join(map(str, P)) + ')') for P in ratpts['rat_pts'])
data['rat_pts_v'] = 2 if ratpts['rat_pts_v'] else 1
# data['mw_rank'] = ratpts['mw_rank']
# data['mw_rank_v'] = ratpts['mw_rank_v']
else:
data['rat_pts_v'] = 0
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl (curve['two_torsion_field'][0], field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data['two_torsion_field_knowl'] = """splitting field of \(%s\) with Galois group %s"""%(intlist_to_poly(t[1]),group_display_knowl(t[2][0],t[2][1]))
else:
# invariants specific to isogeny class
curves_data = list(db.g2c_curves.search({"class" : curve['class']}, ['label','eqn']))
if not curves_data:
raise KeyError("No curves found in database for isogeny class %s of genus 2 curve %s." %(curve['class'],curve['label']))
data['curves'] = [ {"label" : c['label'], "equation_formatted" : list_to_min_eqn(literal_eval(c['eqn'])), "url": url_for_curve_label(c['label'])} for c in curves_data ]
lfunc_data = db.lfunc_lfunctions.lucky({'Lhash':str(curve['Lhash'])})
if not lfunc_data:
raise KeyError("No Lfunction found in database for isogeny class of genus 2 curve %s." %curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [[nth_prime(n+1),lfunc_data['euler_factors'][n]] for n in range(len(lfunc_data['euler_factors'])) if nth_prime(n+1) < 30 and (data['cond'] % nth_prime(n+1))]
data['good_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['good_lfactors']]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = """Endomorphism %s over \(\Q\):<br>""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_base'], endo['factorsRR_base'], ring=data['end_ring_base'] if is_curve else None)
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = """Endomorphism %s over \(\overline{\Q}\):""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_geom'], data['factorsRR_geom'], field=r'\overline{\Q}', ring=data['end_ring_geom'] if is_curve else None)
data['real_geom_end_alg_name'] = real_geom_end_alg_name(curve['real_geom_end_alg'])
data['geom_end_alg_name'] = geom_end_alg_name(curve['geom_end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(data['is_simple_geom'], data['split_field_label'], data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'], data.get('split_labels'), data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
self.plot = encode_plot(eqn_list_to_curve_plot(data['min_eqn'], data['rat_pts'].split(',') if 'rat_pts' in data else []))
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(self.plot)
properties += [
(None, plot_link),
('Conductor',str(data['cond'])),
('Discriminant', str(data['disc'])),
]
properties += [
('Sato-Tate group', data['st_group_link']),
('\(\\End(J_{\\overline{\\Q}}) \\otimes \\R\)', '\(%s\)' % data['real_geom_end_alg_name']),
('\(\\End(J_{\\overline{\\Q}}) \\otimes \\Q\)', '\(%s\)' % data['geom_end_alg_name']),
('\(\\overline{\\Q}\)-simple', bool_pretty(data['is_simple_geom'])),
('\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = []
if is_curve:
friends.append(('Isogeny class %s.%s' % (data['slabel'][0], data['slabel'][1]), url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1])))
# first deal with EC
ecs = []
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
ecs.append(("Elliptic curve " + friend_label, url_for_ec(friend_label)))
else:
ecs.append(("Isogeny class " + ec_label_class(friend_label), url_for_ec_class(friend_label)))
ecs.sort(key=lambda x: key_for_numerically_sort(x[0]))
# then again EC from lfun
instances = []
for elt in db.lfunc_instances.search({'Lhash':data['Lhash'], 'type' : 'ECQP'}, 'url'):
instances.extend(elt.split('|'))
# and then the other isogeny friends
instances.extend([
elt['url'] for elt in
get_instances_by_Lhash_and_trace_hash(data["Lhash"],
4,
int(data["Lhash"])
)
])
exclude = {elt[1].rstrip('/').lstrip('/') for elt in self.friends
if elt[1]}
exclude.add(data['lfunc_url'].lstrip('/L/').rstrip('/'))
for elt in ecs + names_and_urls(instances, exclude=exclude):
# because of the splitting we must use G2C specific code
add_friend(friends, elt)
if is_curve:
friends.append(('Twists', url_for(".index_Q",
g20=str(data['g2'][0]),
g21=str(data['g2'][1]),
g22=str(data['g2'][2]))))
friends.append(('L-function', data['lfunc_url']))
# Breadcrumbs
self.bread = bread = [
('Genus 2 Curves', url_for(".index")),
('$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0], url_for(".by_conductor", cond=data['slabel'][0])),
('%s' % data['slabel'][1], url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1]))
]
if is_curve:
bread += [
('%s' % data['slabel'][2], url_for(".by_url_isogeny_class_discriminant", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2])),
('%s' % data['slabel'][3], url_for(".by_url_curve_label", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2], num=data['slabel'][3]))
]
# Title
self.title = "Genus 2 " + ("Curve " if is_curve else "Isogeny Class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage':'','magma':''} # use default show names
code['curve'] = {'sage':'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s))'%(data['min_eqn'][0],data['min_eqn'][1]),
'magma':'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'%(data['min_eqn'][0],data['min_eqn'][1])}
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation('+str(data['cond'])+',2),R!'+str(bad2)+'>*]'
else:
magma_cond_option = ''
code['cond'] = {'magma': 'Conductor(LSeries(C%s)); Factorization($1);'% magma_cond_option}
code['disc'] = {'magma':'Discriminant(C); Factorization(Integers()!$1);'}
code['igusa_clebsch'] = {'sage':'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':'IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];'}
code['igusa'] = {'magma':'IgusaInvariants(C); [Factorization(Integers()!a): a in $1];'}
code['g2'] = {'magma':'G2Invariants(C);'}
code['aut'] = {'magma':'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {'magma':'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'}
code['num_rat_wpts'] = {'magma':'#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'}
if ratpts:
code['rat_pts'] = {'magma': '[' + ','.join(["C![%s,%s,%s]"%(p[0],p[1],p[2]) for p in ratpts['rat_pts']]) + '];' }
code['two_selmer'] = {'magma':'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'}
code['has_square_sha'] = {'magma':'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {'magma':'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'}
code['torsion_subgroup'] = {'magma':'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'}