def allgens(line): r""" Parses one line from an allgens file. Returns the label and a dict containing fields with keys 'conductor', 'iso', 'number', 'ainvs', 'jinv', 'cm', 'rank', 'gens', 'torsion_order', 'torsion_structure', 'torsion_generators', all values being strings or ints. Input line fields: conductor iso number ainvs rank torsion_structure gens torsion_gens Sample input line: 20202 i 2 [1,0,0,-298389,54947169] 1 [2,4] [-570:6603:1] [-622:311:1] [834:19239:1] """ data = split(line) label = data[0] + data[1] + data[2] rank = int(data[4]) t = data[5] if t == '[]': t = [] else: t = [int(c) for c in t[1:-1].split(",")] torsion = int(prod([ti for ti in t], 1)) ainvs = parse_ainvs(data[3]) E = EllipticCurve([ZZ(a) for a in ainvs]) jinv = unicode(str(E.j_invariant())) if E.has_cm(): cm = int(E.cm_discriminant()) else: cm = int(0) content = { 'conductor': int(data[0]), 'iso': data[0] + data[1], 'number': int(data[2]), 'ainvs': ainvs, 'jinv': jinv, 'cm': cm, 'rank': int(data[4]), 'gens': ["(%s)" % gen[1:-1] for gen in data[6:6 + rank]], 'torsion': torsion, 'torsion_structure': ["%s" % tor for tor in t], 'torsion_generators': ["%s" % parse_tgens(tgens[1:-1]) for tgens in data[6 + rank:]], } extra_data = make_extra_data(label, content['number'], ainvs, content['gens']) content.update(extra_data) return label, content
def allgens(line): r""" Parses one line from an allgens file. Returns the label and a dict containing fields with keys 'conductor', 'iso', 'number', 'ainvs', 'jinv', 'cm', 'rank', 'gens', 'torsion_order', 'torsion_structure', 'torsion_generators', all values being strings or ints. Input line fields: conductor iso number ainvs rank torsion_structure gens torsion_gens Sample input line: 20202 i 2 [1,0,0,-298389,54947169] 1 [2,4] [-570:6603:1] [-622:311:1] [834:19239:1] """ data = split(line) label = data[0] + data[1] + data[2] rank = int(data[4]) t = data[5] if t=='[]': t = [] else: t = [int(c) for c in t[1:-1].split(",")] torsion = int(prod([ti for ti in t], 1)) ainvs = parse_ainvs(data[3]) E = EllipticCurve([ZZ(a) for a in ainvs]) jinv = unicode(str(E.j_invariant())) if E.has_cm(): cm = int(E.cm_discriminant()) else: cm = int(0) content = { 'conductor': int(data[0]), 'iso': data[0] + data[1], 'number': int(data[2]), 'ainvs': ainvs, 'jinv': jinv, 'cm': cm, 'rank': int(data[4]), 'gens': ["(%s)" % gen[1:-1] for gen in data[6:6 + rank]], 'torsion': torsion, 'torsion_structure': ["%s" % tor for tor in t], 'torsion_generators': ["%s" % parse_tgens(tgens[1:-1]) for tgens in data[6 + rank:]], } extra_data = make_extra_data(label,content['number'],ainvs,content['gens']) content.update(extra_data) return label, content
def render_curve_webpage_by_label(label): C = lmfdb.base.getDBConnection() data = C.elliptic_curves.curves.find_one({'lmfdb_label': label}) if data is None: return elliptic_curve_jump_error(label, {}) info = {} ainvs = [int(a) for a in data['ainvs']] E = EllipticCurve(ainvs) cremona_label = data['label'] lmfdb_label = data['lmfdb_label'] N = ZZ(data['conductor']) cremona_iso_class = data['iso'] # eg '37a' lmfdb_iso_class = data['lmfdb_iso'] # eg '37.a' rank = data['rank'] try: j_invariant = QQ(str(data['jinv'])) except KeyError: j_invariant = E.j_invariant() if j_invariant == 0: j_inv_factored = latex(0) else: j_inv_factored = latex(j_invariant.factor()) jinv = unicode(str(j_invariant)) CMD = 0 CM = "no" EndE = "\(\Z\)" if E.has_cm(): CMD = E.cm_discriminant() CM = "yes (\(%s\))" % CMD if CMD % 4 == 0: d4 = ZZ(CMD) // 4 # r = d4.squarefree_part() # f = (d4//r).isqrt() # f="" if f==1 else str(f) # EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r) EndE = "\(\Z[\sqrt{%s}]\)" % (d4) else: EndE = "\(\Z[(1+\sqrt{%s})/2]\)" % CMD # plot=E.plot() discriminant = E.discriminant() xintpoints_projective = [ E.lift_x(x) for x in xintegral_point(data['x-coordinates_of_integral_points']) ] xintpoints = proj_to_aff(xintpoints_projective) if 'degree' in data: modular_degree = data['degree'] else: try: modular_degree = E.modular_degree() except RuntimeError: modular_degree = 0 # invalid, will be displayed nicely G = E.torsion_subgroup().gens() E_pari = E.pari_curve(prec=200) from sage.libs.pari.all import PariError try: minq = E.minimal_quadratic_twist()[0] except PariError: # this does occur with 164411a1 print "PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label minq = E if E == minq: minq_label = lmfdb_label else: minq_ainvs = [str(c) for c in minq.ainvs()] minq_label = C.elliptic_curves.curves.find_one({'ainvs': minq_ainvs })['lmfdb_label'] # We do not just do the following, as Sage's installed database # might not have all the curves in the LMFDB database. # minq_label = E.minimal_quadratic_twist()[0].label() if 'gens' in data: generator = parse_gens(data['gens']) if len(G) == 0: tor_struct = '\mathrm{Trivial}' tor_group = '\mathrm{Trivial}' else: tor_group = ' \\times '.join(['\Z/{%s}\Z' % a.order() for a in G]) if 'torsion_structure' in data: info['tor_structure'] = ' \\times '.join( ['\Z/{%s}\Z' % int(a) for a in data['torsion_structure']]) else: info['tor_structure'] = tor_group def trim_galois_image_code(s): return s[2:] if s[1].isdigit() else s[1:] if 'galois_images' in data: galois_images = data['galois_images'] galois_images = [trim_galois_image_code(s) for s in galois_images] non_surjective_primes = data['non-surjective_primes'] galois_data = [{ 'p': p, 'image': im } for p, im in zip(non_surjective_primes, galois_images)] info.update(data) if rank >= 2: lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}") elif rank == 1: lder_tex = "L%s(E,1)" % ("'" * rank) else: assert rank == 0 lder_tex = "L(E,1)" info['Gamma0optimal'] = (cremona_label[-1] == '1' if cremona_iso_class != '990h' else cremona_label[-1] == '3') info['modular_degree'] = modular_degree p_adic_data_exists = (C.elliptic_curves.padic_db.find({ 'lmfdb_iso': lmfdb_iso_class }).count()) > 0 and info['Gamma0optimal'] # Local data local_data = [] for p in N.prime_factors(): local_info = E.local_data(p, algorithm="generic") local_data.append({ 'p': p, 'tamagawa_number': local_info.tamagawa_number(), 'kodaira_symbol': web_latex(local_info.kodaira_symbol()).replace('$', ''), 'reduction_type': local_info.bad_reduction_type() }) mod_form_iso = lmfdb_label_regex.match(lmfdb_iso_class).groups()[1] tamagawa_numbers = [ E.local_data(p, algorithm="generic").tamagawa_number() for p in N.prime_factors() ] # if we use E.tamagawa_numbers() it calls E.local_data(p) which # crashes on some curves e.g. 164411a1 info.update({ 'conductor': N, 'disc_factor': latex(discriminant.factor()), 'j_invar_factor': j_inv_factored, 'label': lmfdb_label, 'cremona_label': cremona_label, 'iso_class': lmfdb_iso_class, 'cremona_iso_class': cremona_iso_class, 'equation': web_latex(E), #'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250), 'f': web_latex(E.q_eigenform(10)), 'generators': ', '.join(web_latex(g) for g in generator) if 'gens' in data else ' ', 'lder': lder_tex, 'p_adic_primes': [ p for p in sage.all.prime_range(5, 100) if E.is_ordinary(p) and not p.divides(N) ], 'p_adic_data_exists': p_adic_data_exists, 'ainvs': format_ainvs(data['ainvs']), 'CM': CM, 'CMD': CMD, 'EndE': EndE, 'tamagawa_numbers': r' \cdot '.join(str(sage.all.factor(c)) for c in tamagawa_numbers), 'local_data': local_data, 'cond_factor': latex(N.factor()), 'galois_data': galois_data, 'xintegral_points': ', '.join(web_latex(P) for P in xintpoints), 'tor_gens': ', '.join(web_latex(eval(g)) for g in data['torsion_generators']) if False else ', '.join( web_latex(P.element().xy()) for P in list(G)) }) info['friends'] = [('Isogeny class ' + lmfdb_iso_class, url_for(".by_ec_label", label=lmfdb_iso_class)), ('Minimal quadratic twist ' + minq_label, url_for(".by_ec_label", label=minq_label)), ('All twists ', url_for(".rational_elliptic_curves", jinv=jinv)), ('L-function', url_for("l_functions.l_function_ec_page", label=lmfdb_label)), ('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=lmfdb_iso_class)), ('Symmetric 4th power L-function', url_for("l_functions.l_function_ec_sym_page", power='4', label=lmfdb_iso_class))] info['friends'].append( ('Modular form ' + lmfdb_iso_class.replace('.', '.2'), url_for("emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=mod_form_iso))) info['downloads'] = [('Download coeffients of q-expansion', url_for(".download_EC_qexp", label=lmfdb_label, limit=100)), ('Download all stored data', url_for(".download_EC_all", label=lmfdb_label))] # info['learnmore'] = [('Elliptic Curves', url_for(".not_yet_implemented"))] # info['plot'] = image_src(plot) info['plot'] = url_for('.plot_ec', label=lmfdb_label) properties2 = [('Label', '%s' % lmfdb_label), (None, '<img src="%s" width="200" height="150"/>' % url_for('.plot_ec', label=lmfdb_label)), ('Conductor', '\(%s\)' % N), ('Discriminant', '\(%s\)' % discriminant), ('j-invariant', '%s' % web_latex(j_invariant)), ('CM', '%s' % CM), ('Rank', '\(%s\)' % rank), ('Torsion Structure', '\(%s\)' % tor_group)] # properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ]) credit = 'John Cremona and Andrew Sutherland' if info['label'] == info['cremona_label']: t = "Elliptic Curve %s" % info['label'] else: t = "Elliptic Curve %s (Cremona label %s)" % (info['label'], info['cremona_label']) bread = [('Elliptic Curves ', url_for(".rational_elliptic_curves")), ('Elliptic curves %s' % lmfdb_label, ' ')] return render_template("curve.html", properties2=properties2, credit=credit, bread=bread, title=t, info=info, friends=info['friends'], downloads=info['downloads'])
class WebEC(object): """ Class for an elliptic curve over Q """ def __init__(self, dbdata): """ Arguments: - dbdata: the data from the database """ logger.info("Constructing an instance of ECisog_class") self.__dict__.update(dbdata) # Next lines because the hyphens make trouble self.xintcoords = parse_list(dbdata['x-coordinates_of_integral_points']) self.non_surjective_primes = dbdata['non-surjective_primes'] self.make_curve() @staticmethod def by_label(label): """ Searches for a specific elliptic curve in the curves collection by its label, which can be either in LMFDB or Cremona format. """ print "curve label = %s" % label try: N, iso, number = lmfdb_label_regex.match(label).groups() data = db_ec().find_one({"lmfdb_label" : label}) except AttributeError: try: N, iso, number = cremona_label_regex.match(label).groups() data = db_ec().find_one({"label" : label}) except AttributeError: return "Invalid label" # caller must catch this and raise an error if data: return WebEC(data) return "Curve not found" # caller must catch this and raise an error def make_curve(self): # To start with the data fields of self are just those from # the database. We need to reformat these, construct the # actual elliptic curve E, and compute some further (easy) # data about it. # # Weierstrass equation data = self.data = {} data['ainvs'] = [int(ai) for ai in self.ainvs] self.E = EllipticCurve(data['ainvs']) data['equation'] = web_latex(self.E) # conductor, j-invariant and discriminant data['conductor'] = N = ZZ(self.conductor) bad_primes = N.prime_factors() try: data['j_invariant'] = QQ(str(self.jinv)) except KeyError: data['j_invariant'] = self.E.j_invariant() data['j_inv_factor'] = latex(0) if data['j_invariant']: data['j_inv_factor'] = latex(data['j_invariant'].factor()) data['j_inv_str'] = unicode(str(data['j_invariant'])) data['j_inv_latex'] = web_latex(data['j_invariant']) data['disc'] = self.E.discriminant() data['disc_latex'] = web_latex(data['disc']) data['disc_factor'] = latex(data['disc'].factor()) data['cond_factor'] =latex(N.factor()) data['cond_latex'] = web_latex(N) # CM and endomorphism ring data['CMD'] = 0 data['CM'] = "no" data['EndE'] = "\(\Z\)" if self.E.has_cm(): data['CMD'] = self.E.cm_discriminant() data['CM'] = "yes (\(D=%s\))" % data['CMD'] if data['CMD']%4==0: d4 = ZZ(data['CMD'])//4 data['EndE'] = "\(\Z[\sqrt{%s}]\)" % d4 else: data['EndE'] = "\(\Z[(1+\sqrt{%s})/2]\)" % data['CMD'] # modular degree try: data['degree'] = self.degree except AttributeError: try: data['degree'] = self.E.modular_degree() except RuntimeError: data['degree'] # invalid, but will be displayed nicely # Minimal quadratic twist E_pari = self.E.pari_curve(prec=200) from sage.libs.pari.all import PariError try: minq = self.E.minimal_quadratic_twist()[0] except PariError: # this does occur with 164411a1 ec.debug("PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label) minq = self.E if self.E == minq: data['minq_label'] = self.lmfdb_label else: minq_ainvs = [str(c) for c in minq.ainvs()] data['minq_label'] = db_ec().find_one({'ainvs': minq_ainvs})['lmfdb_label'] # rational and integral points mw = self.mw = {} xintpoints_projective = [self.E.lift_x(x) for x in self.xintcoords] xintpoints = [P.xy() for P in xintpoints_projective] mw['int_points'] = ', '.join(web_latex(P) for P in xintpoints) # Generators of infinite order mw['rank'] = self.rank try: mw['generators'] = ', '.join(web_latex(self.E(g).xy()) for g in parse_points(self.gens)) except AttributeError: mw['generators'] = '' # Torsion subgroup: order, structure, generators mw['tor_order'] = self.torsion tor_struct = [int(c) for c in self.torsion_structure] if mw['tor_order'] == 1: mw['tor_struct'] = '\mathrm{Trivial}' mw['tor_gens'] = '' else: mw['tor_struct'] = ' \\times '.join(['\Z/{%s}\Z' % n for n in tor_struct]) mw['tor_gens'] = ', '.join(web_latex(self.E(g).xy()) for g in parse_points(self.torsion_generators)) # Images of Galois representations try: data['galois_images'] = [trim_galois_image_code(s) for s in self.galois_images] data['non_surjective_primes'] = self.non_surjective_primes except AttributeError: print "No Galois image data" data['galois_images'] = [] data['non_surjective_primes'] = [] data['galois_data'] = [{'p': p,'image': im } for p,im in zip(data['non_surjective_primes'], data['galois_images'])] # Leading term of L-function & BSD data bsd = self.bsd = {} if mw['rank'] >= 2: bsd['lder_name'] = "L^{(%s)}(E,1)" % mw['rank'] elif mw['rank']: bsd['lder_name'] = "L'(E,1)" else: bsd['lder_name'] = "L(E,1)" bsd['reg'] = self.regulator bsd['omega'] = self.real_period bsd['sha'] = int(0.1+self.sha_an) bsd['lder'] = self.special_value # Optimality (the optimal curve in the class is the curve # whose Cremona label ends in '1' except for '990h' which was # labelled wrongly long ago) if self.iso == '990h': data['Gamma0optimal'] = bool(self.number == 3) else: data['Gamma0optimal'] = bool(self.number == 1) data['p_adic_data_exists'] = False if data['Gamma0optimal']: data['p_adic_data_exists'] = (padic_db().find({'lmfdb_iso': self.lmfdb_iso}).count()) > 0 data['p_adic_primes'] = [p for p in sage.all.prime_range(5, 100) if self.E.is_ordinary(p) and not p.divides(N)] # Local data local_data = self.local_data = [] # if we use E.tamagawa_numbers() it calls E.local_data(p) which # crashes on some curves e.g. 164411a1 tamagawa_numbers = [] for p in bad_primes: local_info = self.E.local_data(p, algorithm="generic") local_data_p = {} local_data_p['p'] = p local_data_p['tamagawa_number'] = local_info.tamagawa_number() tamagawa_numbers.append(ZZ(local_info.tamagawa_number())) local_data_p['kodaira_symbol'] = web_latex(local_info.kodaira_symbol()).replace('$', '') local_data_p['reduction_type'] = local_info.bad_reduction_type() local_data.append(local_data_p) bsd['tamagawa_factors'] = r' \cdot '.join(str(c.factor()) for c in tamagawa_numbers) bsd['tamagawa_product'] = sage.misc.all.prod(tamagawa_numbers) mod_form_iso = lmfdb_label_regex.match(self.lmfdb_iso).groups()[1] data['newform'] = web_latex(self.E.q_eigenform(10)) self.friends = [ ('Isogeny class ' + self.lmfdb_iso, url_for(".by_ec_label", label=self.lmfdb_iso)), ('Minimal quadratic twist ' + data['minq_label'], url_for(".by_ec_label", label=data['minq_label'])), ('All twists ', url_for(".rational_elliptic_curves", jinv=self.jinv)), ('L-function', url_for("l_functions.l_function_ec_page", label=self.lmfdb_label)), ('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=self.lmfdb_iso)), ('Symmetric 4th power L-function', url_for("l_functions.l_function_ec_sym_page", power='4', label=self.lmfdb_iso)), ('Modular form ' + self.lmfdb_iso.replace('.', '.2'), url_for("emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=mod_form_iso))] self.downloads = [('Download coeffients of q-expansion', url_for(".download_EC_qexp", label=self.lmfdb_label, limit=100)), ('Download all stored data', url_for(".download_EC_all", label=self.lmfdb_label))] self.plot = encode_plot(self.E.plot()) self.plot_link = '<img src="%s" width="200" height="150"/>' % self.plot self.properties = [('Label', self.lmfdb_label), (None, self.plot_link), ('Conductor', '\(%s\)' % data['conductor']), ('Discriminant', '\(%s\)' % data['disc']), ('j-invariant', '%s' % data['j_inv_latex']), ('CM', '%s' % data['CM']), ('Rank', '\(%s\)' % mw['rank']), ('Torsion Structure', '\(%s\)' % mw['tor_struct']) ] if self.lmfdb_iso == self.iso: self.title = "Elliptic Curve %s" % self.lmfdb_label else: self.title = "Elliptic Curve %s (Cremona label %s)" % (self.lmfdb_label, self.label) self.bread = [('Elliptic Curves ', url_for(".rational_elliptic_curves")), ('isogeny class %s' % self.lmfdb_iso, ' ')]
def render_curve_webpage_by_label(label): C = lmfdb.base.getDBConnection() data = C.elliptic_curves.curves.find_one({'lmfdb_label': label}) if data is None: return elliptic_curve_jump_error(label, {}) info = {} ainvs = [int(a) for a in data['ainvs']] E = EllipticCurve(ainvs) cremona_label = data['label'] lmfdb_label = data['lmfdb_label'] N = ZZ(data['conductor']) cremona_iso_class = data['iso'] # eg '37a' lmfdb_iso_class = data['lmfdb_iso'] # eg '37.a' rank = data['rank'] try: j_invariant = QQ(str(data['jinv'])) except KeyError: j_invariant = E.j_invariant() if j_invariant == 0: j_inv_factored = latex(0) else: j_inv_factored = latex(j_invariant.factor()) jinv = unicode(str(j_invariant)) CMD = 0 CM = "no" EndE = "\(\Z\)" if E.has_cm(): CMD = E.cm_discriminant() CM = "yes (\(%s\))"%CMD if CMD%4==0: d4 = ZZ(CMD)//4 # r = d4.squarefree_part() # f = (d4//r).isqrt() # f="" if f==1 else str(f) # EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r) EndE = "\(\Z[\sqrt{%s}]\)"%(d4) else: EndE = "\(\Z[(1+\sqrt{%s})/2]\)"%CMD # plot=E.plot() discriminant = E.discriminant() xintpoints_projective = [E.lift_x(x) for x in xintegral_point(data['x-coordinates_of_integral_points'])] xintpoints = proj_to_aff(xintpoints_projective) if 'degree' in data: modular_degree = data['degree'] else: try: modular_degree = E.modular_degree() except RuntimeError: modular_degree = 0 # invalid, will be displayed nicely G = E.torsion_subgroup().gens() minq = E.minimal_quadratic_twist()[0] if E == minq: minq_label = lmfdb_label else: minq_ainvs = [str(c) for c in minq.ainvs()] minq_label = C.elliptic_curves.curves.find_one({'ainvs': minq_ainvs})['lmfdb_label'] # We do not just do the following, as Sage's installed database # might not have all the curves in the LMFDB database. # minq_label = E.minimal_quadratic_twist()[0].label() if 'gens' in data: generator = parse_gens(data['gens']) if len(G) == 0: tor_struct = '\mathrm{Trivial}' tor_group = '\mathrm{Trivial}' else: tor_group = ' \\times '.join(['\Z/{%s}\Z' % a.order() for a in G]) if 'torsion_structure' in data: info['tor_structure'] = ' \\times '.join(['\Z/{%s}\Z' % int(a) for a in data['torsion_structure']]) else: info['tor_structure'] = tor_group info.update(data) if rank >= 2: lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}") elif rank == 1: lder_tex = "L%s(E,1)" % ("'" * rank) else: assert rank == 0 lder_tex = "L(E,1)" info['Gamma0optimal'] = ( cremona_label[-1] == '1' if cremona_iso_class != '990h' else cremona_label[-1] == '3') info['modular_degree'] = modular_degree p_adic_data_exists = (C.elliptic_curves.padic_db.find( {'lmfdb_iso': lmfdb_iso_class}).count()) > 0 and info['Gamma0optimal'] # Local data local_data = [] for p in N.prime_factors(): local_info = E.local_data(p) local_data.append({'p': p, 'tamagawa_number': local_info.tamagawa_number(), 'kodaira_symbol': web_latex(local_info.kodaira_symbol()).replace('$', ''), 'reduction_type': local_info.bad_reduction_type() }) mod_form_iso = lmfdb_label_regex.match(lmfdb_iso_class).groups()[1] info.update({ 'conductor': N, 'disc_factor': latex(discriminant.factor()), 'j_invar_factor': j_inv_factored, 'label': lmfdb_label, 'cremona_label': cremona_label, 'iso_class': lmfdb_iso_class, 'cremona_iso_class': cremona_iso_class, 'equation': web_latex(E), #'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250), 'f': web_latex(E.q_eigenform(10)), 'generators': ', '.join(web_latex(g) for g in generator) if 'gens' in data else ' ', 'lder': lder_tex, 'p_adic_primes': [p for p in sage.all.prime_range(5, 100) if E.is_ordinary(p) and not p.divides(N)], 'p_adic_data_exists': p_adic_data_exists, 'ainvs': format_ainvs(data['ainvs']), 'CM': CM, 'CMD': CMD, 'EndE': EndE, 'tamagawa_numbers': r' \cdot '.join(str(sage.all.factor(c)) for c in E.tamagawa_numbers()), 'local_data': local_data, 'cond_factor': latex(N.factor()), 'xintegral_points': ', '.join(web_latex(P) for P in xintpoints), 'tor_gens': ', '.join(web_latex(eval(g)) for g in data['torsion_generators']) if False else ', '.join(web_latex(P.element().xy()) for P in list(G)) }) info['friends'] = [ ('Isogeny class ' + lmfdb_iso_class, "/EllipticCurve/Q/%s" % lmfdb_iso_class), ('Minimal quadratic twist ' + minq_label, "/EllipticCurve/Q/%s" % minq_label), ('All twists ', url_for("rational_elliptic_curves", jinv=jinv)), ('L-function', url_for("l_functions.l_function_ec_page", label=lmfdb_label)), ('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=lmfdb_iso_class)), ('Symmetric 4th power L-function', url_for("l_functions.l_function_ec_sym_page", power='4', label=lmfdb_iso_class))] info['friends'].append(('Modular form ' + lmfdb_iso_class.replace('.', '.2'), url_for( "emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=mod_form_iso))) info['downloads'] = [('Download coeffients of q-expansion', url_for("download_EC_qexp", label=lmfdb_label, limit=100)), ('Download all stored data', url_for("download_EC_all", label=lmfdb_label))] # info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))] # info['plot'] = image_src(plot) info['plot'] = url_for('plot_ec', label=lmfdb_label) properties2 = [('Label', '%s' % lmfdb_label), (None, '<img src="%s" width="200" height="150"/>' % url_for( 'plot_ec', label=lmfdb_label)), ('Conductor', '\(%s\)' % N), ('Discriminant', '\(%s\)' % discriminant), ('j-invariant', '%s' % web_latex(j_invariant)), ('CM', '%s' % CM), ('Rank', '\(%s\)' % rank), ('Torsion Structure', '\(%s\)' % tor_group) ] # properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ]) credit = 'John Cremona' if info['label'] == info['cremona_label']: t = "Elliptic Curve %s" % info['label'] else: t = "Elliptic Curve %s (Cremona label %s)" % (info['label'], info['cremona_label']) bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")), ('Elliptic curves %s' % lmfdb_label, ' ')] return render_template("elliptic_curve/elliptic_curve.html", properties2=properties2, credit=credit, bread=bread, title=t, info=info, friends=info['friends'], downloads=info['downloads'])
def allgens(line): r""" Parses one line from an allgens file. Returns the label and a dict containing fields with keys 'conductor', 'iso', 'number', 'ainvs', 'jinv', 'cm', 'rank', 'gens', 'torsion_order', 'torsion_structure', 'torsion_generators', all values being strings or ints, and more. Input line fields: conductor iso number ainvs rank torsion_structure gens torsion_gens Sample input line: 20202 i 2 [1,0,0,-298389,54947169] 1 [2,4] [-570:6603:1] [-622:311:1] [834:19239:1] """ global lmfdb_label_to_label global label_to_lmfdb_label data = split(line) iso = data[0] + data[1] label = iso + data[2] try: lmfdb_label = label_to_lmfdb_label[label] except AttributeError: print("Label {} not found in label_to_lmfdb_label dict!".format(label)) lmfdb_label = "" global nallgens nallgens += 1 if nallgens % 100 == 0: print("processing allgens for {} (#{})".format(label, nallgens)) rank = int(data[4]) t = data[5] tor_struct = [] if t == '[]' else t[1:-1].split(",") torsion = int(prod([int(ti) for ti in tor_struct], 1)) ainvs = parse_ainvs(data[3]) E = EllipticCurve(ainvs) jinv = text_type(E.j_invariant()) if E.has_cm(): cm = int(E.cm_discriminant()) else: cm = int(0) N = E.conductor() bad_p = N.prime_factors() # will be sorted num_bad_p = len(bad_p) local_data = [{ 'p': int(ld.prime().gen()), 'ord_cond': int(ld.conductor_valuation()), 'ord_disc': int(ld.discriminant_valuation()), 'ord_den_j': int(max(0, -(E.j_invariant().valuation(ld.prime().gen())))), 'red': int(ld.bad_reduction_type()), 'rootno': int(E.root_number(ld.prime().gen())), 'kod': web_latex(ld.kodaira_symbol()).replace('$', ''), 'cp': int(ld.tamagawa_number()) } for ld in E.local_data()] semistable = all([ld['ord_cond'] == 1 for ld in local_data]) gens = [ gen.replace("[", "(").replace("]", ")") for gen in data[6:6 + rank] ] tor_gens = ["%s" % parse_tgens(tgens[1:-1]) for tgens in data[6 + rank:]] from lmfdb.elliptic_curves.web_ec import parse_points heights = [float(E(P).height()) for P in parse_points(gens)] Etw, Dtw = E.minimal_quadratic_twist() if Etw.conductor() == N: min_quad_twist = { 'label': label, 'lmfdb_label': lmfdb_label, 'disc': int(1) } else: minq_ainvs = Etw.ainvs() r = curves.lucky({ 'jinv': str(E.j_invariant()), 'ainvs': minq_ainvs }, projection=['label', 'lmfdb_label']) min_quad_twist = { 'label': r['label'], 'lmfdb_label': r['lmfdb_label'], 'disc': int(Dtw) } trace_hash = TraceHashClass(iso, E) return label, { 'conductor': int(data[0]), 'iso': iso, 'number': int(data[2]), 'ainvs': ainvs, 'jinv': jinv, 'cm': cm, 'rank': rank, 'gens': gens, 'torsion': torsion, 'torsion_structure': tor_struct, 'torsion_generators': tor_gens, 'trace_hash': trace_hash, 'equation': web_latex(E), 'bad_primes': bad_p, 'num_bad_primes': num_bad_p, 'local_data': local_data, 'semistable': semistable, 'signD': int(E.discriminant().sign()), 'heights': heights, 'aplist': E.aplist(100, python_ints=True), 'anlist': E.anlist(20, python_ints=True), 'min_quad_twist': min_quad_twist, }
def render_curve_webpage_by_label(label): C = lmfdb.base.getDBConnection() data = C.elliptic_curves.curves.find_one({"lmfdb_label": label}) if data is None: return elliptic_curve_jump_error(label, {}) info = {} ainvs = [int(a) for a in data["ainvs"]] E = EllipticCurve(ainvs) cremona_label = data["label"] lmfdb_label = data["lmfdb_label"] N = ZZ(data["conductor"]) cremona_iso_class = data["iso"] # eg '37a' lmfdb_iso_class = data["lmfdb_iso"] # eg '37.a' rank = data["rank"] try: j_invariant = QQ(str(data["jinv"])) except KeyError: j_invariant = E.j_invariant() if j_invariant == 0: j_inv_factored = latex(0) else: j_inv_factored = latex(j_invariant.factor()) jinv = unicode(str(j_invariant)) CMD = 0 CM = "no" EndE = "\(\Z\)" if E.has_cm(): CMD = E.cm_discriminant() CM = "yes (\(%s\))" % CMD if CMD % 4 == 0: d4 = ZZ(CMD) // 4 # r = d4.squarefree_part() # f = (d4//r).isqrt() # f="" if f==1 else str(f) # EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r) EndE = "\(\Z[\sqrt{%s}]\)" % (d4) else: EndE = "\(\Z[(1+\sqrt{%s})/2]\)" % CMD # plot=E.plot() discriminant = E.discriminant() xintpoints_projective = [E.lift_x(x) for x in xintegral_point(data["x-coordinates_of_integral_points"])] xintpoints = proj_to_aff(xintpoints_projective) if "degree" in data: modular_degree = data["degree"] else: try: modular_degree = E.modular_degree() except RuntimeError: modular_degree = 0 # invalid, will be displayed nicely G = E.torsion_subgroup().gens() E_pari = E.pari_curve(prec=200) from sage.libs.pari.all import PariError try: minq = E.minimal_quadratic_twist()[0] except PariError: # this does occur with 164411a1 print "PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label minq = E if E == minq: minq_label = lmfdb_label else: minq_ainvs = [str(c) for c in minq.ainvs()] minq_label = C.elliptic_curves.curves.find_one({"ainvs": minq_ainvs})["lmfdb_label"] # We do not just do the following, as Sage's installed database # might not have all the curves in the LMFDB database. # minq_label = E.minimal_quadratic_twist()[0].label() if "gens" in data: generator = parse_gens(data["gens"]) if len(G) == 0: tor_struct = "\mathrm{Trivial}" tor_group = "\mathrm{Trivial}" else: tor_group = " \\times ".join(["\Z/{%s}\Z" % a.order() for a in G]) if "torsion_structure" in data: info["tor_structure"] = " \\times ".join(["\Z/{%s}\Z" % int(a) for a in data["torsion_structure"]]) else: info["tor_structure"] = tor_group info.update(data) if rank >= 2: lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}") elif rank == 1: lder_tex = "L%s(E,1)" % ("'" * rank) else: assert rank == 0 lder_tex = "L(E,1)" info["Gamma0optimal"] = cremona_label[-1] == "1" if cremona_iso_class != "990h" else cremona_label[-1] == "3" info["modular_degree"] = modular_degree p_adic_data_exists = (C.elliptic_curves.padic_db.find({"lmfdb_iso": lmfdb_iso_class}).count()) > 0 and info[ "Gamma0optimal" ] # Local data local_data = [] for p in N.prime_factors(): local_info = E.local_data(p, algorithm="generic") local_data.append( { "p": p, "tamagawa_number": local_info.tamagawa_number(), "kodaira_symbol": web_latex(local_info.kodaira_symbol()).replace("$", ""), "reduction_type": local_info.bad_reduction_type(), } ) mod_form_iso = lmfdb_label_regex.match(lmfdb_iso_class).groups()[1] tamagawa_numbers = [E.local_data(p, algorithm="generic").tamagawa_number() for p in N.prime_factors()] # if we use E.tamagawa_numbers() it calls E.local_data(p) which # crashes on some curves e.g. 164411a1 info.update( { "conductor": N, "disc_factor": latex(discriminant.factor()), "j_invar_factor": j_inv_factored, "label": lmfdb_label, "cremona_label": cremona_label, "iso_class": lmfdb_iso_class, "cremona_iso_class": cremona_iso_class, "equation": web_latex(E), #'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250), "f": web_latex(E.q_eigenform(10)), "generators": ", ".join(web_latex(g) for g in generator) if "gens" in data else " ", "lder": lder_tex, "p_adic_primes": [p for p in sage.all.prime_range(5, 100) if E.is_ordinary(p) and not p.divides(N)], "p_adic_data_exists": p_adic_data_exists, "ainvs": format_ainvs(data["ainvs"]), "CM": CM, "CMD": CMD, "EndE": EndE, "tamagawa_numbers": r" \cdot ".join(str(sage.all.factor(c)) for c in tamagawa_numbers), "local_data": local_data, "cond_factor": latex(N.factor()), "xintegral_points": ", ".join(web_latex(P) for P in xintpoints), "tor_gens": ", ".join(web_latex(eval(g)) for g in data["torsion_generators"]) if False else ", ".join(web_latex(P.element().xy()) for P in list(G)), } ) info["friends"] = [ ("Isogeny class " + lmfdb_iso_class, "/EllipticCurve/Q/%s" % lmfdb_iso_class), ("Minimal quadratic twist " + minq_label, "/EllipticCurve/Q/%s" % minq_label), ("All twists ", url_for("rational_elliptic_curves", jinv=jinv)), ("L-function", url_for("l_functions.l_function_ec_page", label=lmfdb_label)), ( "Symmetric square L-function", url_for("l_functions.l_function_ec_sym_page", power="2", label=lmfdb_iso_class), ), ( "Symmetric 4th power L-function", url_for("l_functions.l_function_ec_sym_page", power="4", label=lmfdb_iso_class), ), ] info["friends"].append( ( "Modular form " + lmfdb_iso_class.replace(".", ".2"), url_for("emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=mod_form_iso), ) ) info["downloads"] = [ ("Download coeffients of q-expansion", url_for("download_EC_qexp", label=lmfdb_label, limit=100)), ("Download all stored data", url_for("download_EC_all", label=lmfdb_label)), ] # info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))] # info['plot'] = image_src(plot) info["plot"] = url_for("plot_ec", label=lmfdb_label) properties2 = [ ("Label", "%s" % lmfdb_label), (None, '<img src="%s" width="200" height="150"/>' % url_for("plot_ec", label=lmfdb_label)), ("Conductor", "\(%s\)" % N), ("Discriminant", "\(%s\)" % discriminant), ("j-invariant", "%s" % web_latex(j_invariant)), ("CM", "%s" % CM), ("Rank", "\(%s\)" % rank), ("Torsion Structure", "\(%s\)" % tor_group), ] # properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ]) credit = "John Cremona" if info["label"] == info["cremona_label"]: t = "Elliptic Curve %s" % info["label"] else: t = "Elliptic Curve %s (Cremona label %s)" % (info["label"], info["cremona_label"]) bread = [("Elliptic Curves ", url_for("rational_elliptic_curves")), ("Elliptic curves %s" % lmfdb_label, " ")] return render_template( "elliptic_curve/elliptic_curve.html", properties2=properties2, credit=credit, bread=bread, title=t, info=info, friends=info["friends"], downloads=info["downloads"], )