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'] = 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['disc_key'][3:]) # use disc_key rather than abs_disc (will work when abs_disc > 2^63) data['disc'] = curve['disc_sign'] * curve['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 i in range(tama.count()): item = tama.next() if item['tamagawa_number'] > 0: tamgwnr = str(item['tamagawa_number']) else: tamgwnr = 'N/A' data['tama'] += tamgwnr + ' (p = ' + str(item['p']) + ')' if (i+1 < tama.count()): data['tama'] += ', ' if ratpts: if len(ratpts['rat_pts']): data['rat_pts'] = ', '.join(web_latex('(' +' : '.join(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], getDBConnection(), 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],getDBConnection())) else: # invariants specific to isogeny class curves_data = g2c_db_curves().find({"class" : curve['class']},{'_id':int(0),'label':int(1),'eqn':int(1),'disc_key':int(1)}).sort([("disc_key", ASCENDING), ("label", ASCENDING)]) 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 = g2c_db_lfunction_by_hash(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'] = end_alg_name(curve['real_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']), ('\(\\overline{\\Q}\)-simple', bool_pretty(data['is_simple_geom'])), ('\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])), ] # Friends self.friends = friends = [('L-function', data['lfunc_url'])] 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]))) for friend in g2c_db_lfunction_instances().find({'Lhash':data['Lhash']},{'_id':False,'url':True}): if 'url' in friend: add_friend (friends, lfunction_friend_from_url(friend['url'])) if 'urls' in friend: for url in friends['urls']: add_friend (friends, lfunction_friend_from_url(friend['url'])) if 'split_labels' in data: for friend_label in data['split_labels']: if is_curve: add_friend (friends, ("Elliptic curve " + friend_label, url_for_ec(friend_label))) else: add_friend (friends, ("EC isogeny class " + ec_label_class(friend_label), url_for_ec_class(friend_label))) 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])))) # 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);'}
def make_object(self, curve, endo, 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'] = 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['disc_key'][3:] ) # use disc_key rather than abs_disc (will work when abs_disc > 2^63) data['disc'] = curve['disc_sign'] * curve['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" data['locally_solvable'] = "yes" if curve[ 'locally_solvable'] else "no" 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'] else: # invariants specific to isogeny class curves_data = g2c_db_curves().find({ "class": curve['class'] }, { '_id': int(0), 'label': int(1), 'eqn': int(1), 'disc_key': int(1) }).sort([("disc_key", ASCENDING), ("label", ASCENDING)]) 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 = g2c_db_lfunction_by_hash(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'] = end_alg_name( curve['real_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'])) plot_link = '<img src="%s" width="200" height="150"/>' % 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']), ('\(\\overline{\\Q}\)-simple', bool_pretty(data['is_simple_geom'])), ('\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])), ] # Friends self.friends = friends = [('L-function', data['lfunc_url'])] 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]))) for friend in g2c_db_lfunction_instances().find( {'Lhash': data['Lhash']}, { '_id': False, 'url': True }): if 'url' in friend: add_friend(friends, lfunction_friend_from_url(friend['url'])) if 'urls' in friend: for url in friends['urls']: add_friend(friends, lfunction_friend_from_url(friend['url'])) if 'split_labels' in data: for friend_label in data['split_labels']: if is_curve: add_friend(friends, ("Elliptic curve " + friend_label, url_for_ec(friend_label))) else: add_friend( friends, ("EC isogeny class " + ec_label_class(friend_label), url_for_ec_class(friend_label))) 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])))) # 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)));' } 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; HasPointsLocallyEverywhere(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);' } code['torsion_subgroup'] = { 'magma': 'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);' }
Z = [0,]*len(X) for i in range(len(X)): x = X[i] z = prime_range(x) # Returns primes 0 <= p < x if is_prime(x): # galois.primes() returns 0 <= p <= x z.append(x) Z[i] = [int(zz) for zz in z] d = {"X": X, "Z": Z} save_pickle(d, FOLDER, "primes.pkl") set_seed(SEED + 302) X = [random.randint(1, 1000) for _ in range(20)] Z = [0,]*len(X) for i in range(len(X)): x = X[i] z = nth_prime(x) Z[i] = int(z) d = {"X": X, "Z": Z} save_pickle(d, FOLDER, "kth_prime.pkl") set_seed(SEED + 303) X = [random.randint(1, 100) for _ in range(20)] + [random.randint(100, 1_000_000) for _ in range(20)] Z = [0,]*len(X) for i in range(len(X)): x = X[i] z = previous_prime(x) # Returns z < x if is_prime(x): # galois.prev_prime() returns z <= x z = x Z[i] = int(z) d = {"X": X, "Z": Z} save_pickle(d, FOLDER, "prev_prime.pkl")