def make_extra_data(label, number, ainvs, gens): """Given a curve label (and number, as some data is only stored wih curve number 1 in each class) and its ainvs and gens, returns a dict with which to update the entry. Extra items computed here: 'equation': latex string of curve's equation 'signD': sign of discriminant 'local_data': list of dicts, one item for each bad prime 'min_quad_twist': dict holding curve's min quadratic twist and the twisting discriminant 'heights': list of heights of gens and for curve #1 in a class only: 'aplist': list of a_p for p<100 'anlist': list of a_n for n<=20 """ E = EllipticCurve(parse_ainvs(ainvs)) data = {} # convert from a list of strings to a single string, e.g. from ['0','0','0','1','1'] to '[0,0,0,1,1]' data['equation'] = web_latex(E) data['signD'] = int(E.discriminant().sign()) data['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('$', ''), 'cp': int(ld.tamagawa_number()) } for ld in E.local_data()] Etw, Dtw = E.minimal_quadratic_twist() if Etw.conductor() == E.conductor(): data['min_quad_twist'] = {'label': label, 'disc': int(1)} else: minq_ainvs = ''.join(['['] + [str(c) for c in Etw.ainvs()] + [']']) r = curves.find_one({ 'jinv': str(E.j_invariant()), 'ainvs': minq_ainvs }) minq_label = "" if r is None else r['label'] data['min_quad_twist'] = {'label': minq_label, 'disc': int(Dtw)} from lmfdb.elliptic_curves.web_ec import parse_points gens = [E(g) for g in parse_points(gens)] data['heights'] = [float(P.height()) for P in gens] if number == 1: data['aplist'] = E.aplist(100, python_ints=True) data['anlist'] = E.anlist(20, python_ints=True) return data
def make_extra_data(label, number, ainvs, gens): """ C is a database elliptic curve entry. Returns a dict with which to update the entry. Data fields needed in C already: 'ainvs', 'lmfdb_label', 'gens', 'number' """ E = EllipticCurve([int(a) for a in ainvs]) data = {} # convert from a list of strings to a single string, e.g. from ['0','0','0','1','1'] to '[0,0,0,1,1]' data['xainvs'] = ''.join(['[', ','.join(ainvs), ']']) data['equation'] = web_latex(E) data['signD'] = int(E.discriminant().sign()) data['local_data'] = [{ 'p': int(ld.prime().gen()), 'ord_cond': int(ld.conductor_valuation()), 'ord_disc': int(ld.discriminant_valuation()), 'ord_den_j': int(max(0, -(E.j_invariant().valuation(ld.prime().gen())))), 'red': int(ld.bad_reduction_type()), 'rootno': int(E.root_number(ld.prime().gen())), 'kod': web_latex(ld.kodaira_symbol()).replace('$', ''), 'cp': int(ld.tamagawa_number()) } for ld in E.local_data()] Etw, Dtw = E.minimal_quadratic_twist() if Etw.conductor() == E.conductor(): data['min_quad_twist'] = {'label': label, 'disc': int(1)} else: # Later this should be changed to look for xainvs but now all curves have ainvs minq_ainvs = [str(c) for c in Etw.ainvs()] r = curves.find_one({ 'jinv': str(E.j_invariant()), 'ainvs': minq_ainvs }) minq_label = "" if r is None else r['label'] data['min_quad_twist'] = {'label': minq_label, 'disc': int(Dtw)} from lmfdb.elliptic_curves.web_ec import parse_points gens = [E(g) for g in parse_points(gens)] data['heights'] = [float(P.height()) for P in gens] if number == 1: data['aplist'] = E.aplist(100, python_ints=True) data['anlist'] = E.anlist(20, python_ints=True) return data
def make_extra_data(label,number,ainvs,gens): """Given a curve label (and number, as some data is only stored wih curve number 1 in each class) and its ainvs and gens, returns a dict with which to update the entry. Extra items computed here: 'equation': latex string of curve's equation 'signD': sign of discriminant 'local_data': list of dicts, one item for each bad prime 'min_quad_twist': dict holding curve's min quadratic twist and the twisting discriminant 'heights': list of heights of gens and for curve #1 in a class only: 'aplist': list of a_p for p<100 'anlist': list of a_n for n<=20 """ E = EllipticCurve(parse_ainvs(ainvs)) data = {} # convert from a list of strings to a single string, e.g. from ['0','0','0','1','1'] to '[0,0,0,1,1]' data['equation'] = web_latex(E) data['signD'] = int(E.discriminant().sign()) data['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()] Etw, Dtw = E.minimal_quadratic_twist() if Etw.conductor()==E.conductor(): data['min_quad_twist'] = {'label':label, 'disc':int(1)} else: minq_ainvs = ''.join(['['] + [str(c) for c in Etw.ainvs()] + [']']) r = curves.find_one({'jinv':str(E.j_invariant()), 'ainvs':minq_ainvs}) minq_label = "" if r is None else r['label'] data['min_quad_twist'] = {'label':minq_label, 'disc':int(Dtw)} from lmfdb.elliptic_curves.web_ec import parse_points gens = [E(g) for g in parse_points(gens)] data['heights'] = [float(P.height()) for P in gens] if number==1: data['aplist'] = E.aplist(100,python_ints=True) data['anlist'] = E.anlist(20,python_ints=True) return data
def tidy_ecdb(C): """A rewrite function for tidying up the curves collection, Feb 2018. """ if C['conductor'] < 380000: return C # 1. delete the old redundant 'ainvs' field (we now use 'xainvs' #C.pop('ainvs') # # 2. add local root number if missing ld = C['local_data'] if not 'rootno' in ld[0]: E = EllipticCurve([int(ai) for ai in C['xainvs'][1:-1].split(",")]) for i, ldp in enumerate(ld): ldp['rootno'] = int(E.root_number(ZZ(ldp['p']))) ld[i] = ldp C['local_data'] = ld return C
def tidy_ecdb(C): """A rewrite function for tidying up the curves collection, Feb 2018. """ if C['conductor']<380000: return C # 1. delete the old redundant 'ainvs' field (we now use 'xainvs' #C.pop('ainvs') # # 2. add local root number if missing ld = C['local_data'] if not 'rootno' in ld[0]: E = EllipticCurve([int(ai) for ai in C['xainvs'][1:-1].split(",")]) for i, ldp in enumerate(ld): ldp['rootno'] = int(E.root_number(ZZ(ldp['p']))) ld[i] = ldp C['local_data'] = ld return C
def make_extra_data(label,number,ainvs,gens): """ C is a database elliptic curve entry. Returns a dict with which to update the entry. Data fields needed in C already: 'ainvs', 'lmfdb_label', 'gens', 'number' """ E = EllipticCurve([int(a) for a in ainvs]) data = {} # convert from a list of strings to a single string, e.g. from ['0','0','0','1','1'] to '[0,0,0,1,1]' data['xainvs'] = ''.join(['[',','.join(ainvs),']']) data['equation'] = web_latex(E) data['signD'] = int(E.discriminant().sign()) data['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()] Etw, Dtw = E.minimal_quadratic_twist() if Etw.conductor()==E.conductor(): data['min_quad_twist'] = {'label':label, 'disc':int(1)} else: # Later this should be changed to look for xainvs but now all curves have ainvs minq_ainvs = [str(c) for c in Etw.ainvs()] r = curves.find_one({'jinv':str(E.j_invariant()), 'ainvs':minq_ainvs}) minq_label = "" if r is None else r['label'] data['min_quad_twist'] = {'label':minq_label, 'disc':int(Dtw)} from lmfdb.elliptic_curves.web_ec import parse_points gens = [E(g) for g in parse_points(gens)] data['heights'] = [float(P.height()) for P in gens] if number==1: data['aplist'] = E.aplist(100,python_ints=True) data['anlist'] = E.anlist(20,python_ints=True) return data
class WebEC(object): """ Class for an elliptic curve over Q """ def __init__(self, dbdata): """ Arguments: - dbdata: the data from the database """ logger.debug("Constructing an instance of WebEC") self.__dict__.update(dbdata) # Next lines because the hyphens make trouble self.xintcoords = split_list( dbdata['x-coordinates_of_integral_points']) self.non_surjective_primes = dbdata['non-surjective_primes'] # Next lines because the python identifiers cannot start with 2 self.twoadic_index = dbdata['2adic_index'] self.twoadic_log_level = dbdata['2adic_log_level'] self.twoadic_gens = dbdata['2adic_gens'] self.twoadic_label = dbdata['2adic_label'] # All other fields are handled here 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. """ try: N, iso, number = split_lmfdb_label(label) data = db_ec().find_one({"lmfdb_label": label}) except AttributeError: try: N, iso, number = split_cremona_label(label) 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. # Old version: required constructing the actual elliptic curve # E, and computing some further data about it. # New version (May 2016): extra data fields now in the # database so we do not have to construct the curve or do any # computation with it on the fly. As a failsafe the old way # is still included. data = self.data = {} try: data['ainvs'] = [int(c) for c in self.xainvs[1:-1].split(',')] except AttributeError: data['ainvs'] = [int(ai) for ai in self.ainvs] data['conductor'] = N = ZZ(self.conductor) data['j_invariant'] = QQ(str(self.jinv)) data['j_inv_factor'] = latex(0) if data['j_invariant']: # don't factor 0 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']) mw = self.mw = {} mw['rank'] = self.rank mw['int_points'] = '' if self.xintcoords: a1, a2, a3, a4, a6 = [ZZ(a) for a in data['ainvs']] def lift_x(x): f = ((x + a2) * x + a4) * x + a6 b = (a1 * x + a3) d = (b * b + 4 * f).sqrt() return (x, (-b + d) / 2) mw['int_points'] = ', '.join( web_latex(lift_x(x)) for x in self.xintcoords) mw['generators'] = '' mw['heights'] = [] if self.gens: mw['generators'] = [ web_latex(tuple(P)) for P in parse_points(self.gens) ] 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(tuple(P)) for P in parse_points(self.torsion_generators)) # try to get all the data we need from the database entry (now in self) try: data['equation'] = self.equation local_data = self.local_data D = self.signD * prod( [ld['p']**ld['ord_disc'] for ld in local_data]) data['disc'] = D Nfac = Factorization([(ZZ(ld['p']), ld['ord_cond']) for ld in local_data]) Dfac = Factorization([(ZZ(ld['p']), ld['ord_disc']) for ld in local_data], unit=ZZ(self.signD)) data['minq_D'] = minqD = self.min_quad_twist['disc'] minq_label = self.min_quad_twist['label'] data['minq_label'] = db_ec().find_one( {'label': minq_label}, ['lmfdb_label'])['lmfdb_label'] data['minq_info'] = '(itself)' if minqD == 1 else '(by %s)' % minqD try: data['degree'] = self.degree except AttributeError: data['degree'] = 0 # invalid, but will be displayed nicely mw['heights'] = self.heights if self.number == 1: data['an'] = self.anlist data['ap'] = self.aplist else: r = db_ec().find_one({ 'lmfdb_iso': self.lmfdb_iso, 'number': 1 }, ['anlist', 'aplist']) data['an'] = r['anlist'] data['ap'] = r['aplist'] # otherwise fall back to computing it from the curve except AttributeError: print("Falling back to constructing E") self.E = EllipticCurve(data['ainvs']) data['equation'] = web_latex(self.E) data['disc'] = D = self.E.discriminant() Nfac = N.factor() Dfac = D.factor() bad_primes = [p for p, e in Nfac] try: data['degree'] = self.degree except AttributeError: try: data['degree'] = self.E.modular_degree() except RuntimeError: data['degree'] = 0 # invalid, but will be displayed nicely minq, minqD = self.E.minimal_quadratic_twist() data['minq_D'] = minqD if minqD == 1: data['minq_label'] = self.lmfdb_label data['minq_info'] = '(itself)' else: # This relies on the minimal twist being in the # database, which is true when the database only # contains the Cremona database. It would be a good # idea if, when the database is extended, we ensured # that for any curve included, all twists of smaller # conductor are also included. minq_ainvs = [str(c) for c in minq.ainvs()] data['minq_label'] = db_ec().find_one( { 'jinv': str(self.E.j_invariant()), 'ainvs': minq_ainvs }, ['lmfdb_label'])['lmfdb_label'] data['minq_info'] = '(by %s)' % minqD if self.gens: self.generators = [self.E(g) for g in parse_points(self.gens)] mw['heights'] = [P.height() for P in self.generators] data['an'] = self.E.anlist(20, python_ints=True) data['ap'] = self.E.aplist(100, python_ints=True) self.local_data = local_data = [] for p in bad_primes: ld = self.E.local_data(p, algorithm="generic") local_data_p = {} local_data_p['p'] = p local_data_p['cp'] = ld.tamagawa_number() local_data_p['kod'] = web_latex(ld.kodaira_symbol()).replace( '$', '') local_data_p['red'] = ld.bad_reduction_type() rootno = -ld.bad_reduction_type() if rootno == 0: rootno = self.E.root_number(p) local_data_p['rootno'] = rootno local_data_p['ord_cond'] = ld.conductor_valuation() local_data_p['ord_disc'] = ld.discriminant_valuation() local_data_p['ord_den_j'] = max( 0, -self.E.j_invariant().valuation(p)) local_data.append(local_data_p) # If we got the data from the database, the root numbers may # not have been stored there, so we have to compute them. If # there are additive primes this means constructing the curve. for ld in self.local_data: if not 'rootno' in ld: rootno = -ld['red'] if rootno == 0: try: E = self.E except AttributeError: self.E = E = EllipticCurve(data['ainvs']) rootno = E.root_number(ld['p']) ld['rootno'] = rootno minq_N, minq_iso, minq_number = split_lmfdb_label(data['minq_label']) data['disc_factor'] = latex(Dfac) data['cond_factor'] = latex(Nfac) data['disc_latex'] = web_latex(D) data['cond_latex'] = web_latex(N) data['CMD'] = self.cm data['CM'] = "no" data['EndE'] = "\(\Z\)" if self.cm: 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'] data['ST'] = st_link_by_name(1, 2, 'N(U(1))') else: data['ST'] = st_link_by_name(1, 2, 'SU(2)') data['p_adic_primes'] = [ p for i, p in enumerate(prime_range(5, 100)) if (N * data['ap'][i]) % p != 0 ] 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'])] cond, iso, num = split_lmfdb_label(self.lmfdb_label) self.class_url = url_for(".by_double_iso_label", conductor=N, iso_label=iso) self.ncurves = db_ec().count({'lmfdb_iso': self.lmfdb_iso}) isodegs = [str(d) for d in self.isogeny_degrees if d > 1] if len(isodegs) < 3: data['isogeny_degrees'] = " and ".join(isodegs) else: data['isogeny_degrees'] = " and ".join( [", ".join(isodegs[:-1]), isodegs[-1]]) if self.twoadic_gens: from sage.matrix.all import Matrix data['twoadic_gen_matrices'] = ','.join( [latex(Matrix(2, 2, M)) for M in self.twoadic_gens]) data[ 'twoadic_rouse_url'] = ROUSE_URL_PREFIX + self.twoadic_label + ".html" # Leading term of L-function & BSD data bsd = self.bsd = {} r = self.rank if r >= 2: bsd['lder_name'] = "L^{(%s)}(E,1)/%s!" % (r, r) elif r: 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 tamagawa_numbers = [ZZ(ld['cp']) for ld in local_data] cp_fac = [cp.factor() for cp in tamagawa_numbers] cp_fac = [ latex(cp) if len(cp) < 2 else '(' + latex(cp) + ')' for cp in cp_fac ] bsd['tamagawa_factors'] = r'\cdot'.join(cp_fac) bsd['tamagawa_product'] = prod(tamagawa_numbers) data['newform'] = web_latex( PowerSeriesRing(QQ, 'q')(data['an'], 20, check=True)) data['newform_label'] = self.newform_label = newform_label( cond, 2, 1, iso) self.newform_link = url_for("emf.render_elliptic_modular_forms", level=cond, weight=2, character=1, label=iso) self.newform_exists_in_db = is_newform_in_db(self.newform_label) self._code = None self.class_url = url_for(".by_double_iso_label", conductor=N, iso_label=iso) self.friends = [('Isogeny class ' + self.lmfdb_iso, self.class_url), ('Minimal quadratic twist %s %s' % (data['minq_info'], data['minq_label']), url_for(".by_triple_label", conductor=minq_N, iso_label=minq_iso, number=minq_number)), ('All twists ', url_for(".rational_elliptic_curves", jinv=self.jinv)), ('L-function', url_for("l_functions.l_function_ec_page", label=self.lmfdb_label))] if not self.cm: if N <= 300: self.friends += [('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=self.lmfdb_iso))] if N <= 50: self.friends += [('Symmetric cube L-function', url_for("l_functions.l_function_ec_sym_page", power='3', label=self.lmfdb_iso))] if self.newform_exists_in_db: self.friends += [('Modular form ' + self.newform_label, self.newform_link)] self.downloads = [('Download coefficients of q-expansion', url_for(".download_EC_qexp", label=self.lmfdb_label, limit=1000)), ('Download all stored data', url_for(".download_EC_all", label=self.lmfdb_label)), ('Download Magma code', url_for(".ec_code_download", conductor=cond, iso=iso, number=num, label=self.lmfdb_label, download_type='magma')), ('Download Sage code', url_for(".ec_code_download", conductor=cond, iso=iso, number=num, label=self.lmfdb_label, download_type='sage')), ('Download GP code', url_for(".ec_code_download", conductor=cond, iso=iso, number=num, label=self.lmfdb_label, download_type='gp'))] try: self.plot = encode_plot(self.E.plot()) except AttributeError: self.plot = encode_plot(EllipticCurve(data['ainvs']).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'])] self.title = "Elliptic Curve %s (Cremona label %s)" % ( self.lmfdb_label, self.label) self.bread = [('Elliptic Curves', url_for("ecnf.index")), ('$\Q$', url_for(".rational_elliptic_curves")), ('%s' % N, url_for(".by_conductor", conductor=N)), ('%s' % iso, url_for(".by_double_iso_label", conductor=N, iso_label=iso)), ('%s' % num, ' ')] def code(self): if self._code == None: self.make_code_snippets() return self._code def make_code_snippets(self): # read in code.yaml from current directory: _curdir = os.path.dirname(os.path.abspath(__file__)) self._code = yaml.load(open(os.path.join(_curdir, "code.yaml"))) # Fill in placeholders for this specific curve: for lang in ['sage', 'pari', 'magma']: self._code['curve'][lang] = self._code['curve'][lang] % ( self.data['ainvs'], self.label) return for k in self._code: if k != 'prompt': for lang in self._code[k]: self._code[k][lang] = self._code[k][lang].split("\n") # remove final empty line if len(self._code[k][lang][-1]) == 0: self._code[k][lang] = self._code[k][lang][:-1]
class WebEC(object): """ Class for an elliptic curve over Q """ def __init__(self, dbdata): """ Arguments: - dbdata: the data from the database """ logger.debug("Constructing an instance of ECisog_class") self.__dict__.update(dbdata) # Next lines because the hyphens make trouble self.xintcoords = split_list(dbdata['x-coordinates_of_integral_points']) self.non_surjective_primes = dbdata['non-surjective_primes'] # Next lines because the python identifiers cannot start with 2 self.twoadic_index = dbdata['2adic_index'] self.twoadic_log_level = dbdata['2adic_log_level'] self.twoadic_gens = dbdata['2adic_gens'] self.twoadic_label = dbdata['2adic_label'] # All other fields are handled here 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. """ try: N, iso, number = split_lmfdb_label(label) data = db_ec().find_one({"lmfdb_label" : label}) except AttributeError: try: N, iso, number = split_cremona_label(label) 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. # Old version: required constructing the actual elliptic curve # E, and computing some further data about it. # New version (May 2016): extra data fields now in the # database so we do not have to construct the curve or do any # computation with it on the fly. As a failsafe the old way # is still included. data = self.data = {} try: data['ainvs'] = [int(c) for c in self.xainvs[1:-1].split(',')] except AttributeError: data['ainvs'] = [int(ai) for ai in self.ainvs] data['conductor'] = N = ZZ(self.conductor) data['j_invariant'] = QQ(str(self.jinv)) data['j_inv_factor'] = latex(0) if data['j_invariant']: # don't factor 0 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']) mw = self.mw = {} mw['rank'] = self.rank mw['int_points'] = '' if self.xintcoords: a1, a2, a3, a4, a6 = [ZZ(a) for a in data['ainvs']] def lift_x(x): f = ((x + a2) * x + a4) * x + a6 b = (a1*x + a3) d = (b*b + 4*f).sqrt() return (x, (-b+d)/2) mw['int_points'] = ', '.join(web_latex(lift_x(x)) for x in self.xintcoords) mw['generators'] = '' mw['heights'] = [] if self.gens: mw['generators'] = [web_latex(tuple(P)) for P in parse_points(self.gens)] 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(tuple(P)) for P in parse_points(self.torsion_generators)) # try to get all the data we need from the database entry (now in self) try: data['equation'] = self.equation local_data = self.local_data D = self.signD * prod([ld['p']**ld['ord_disc'] for ld in local_data]) data['disc'] = D Nfac = Factorization([(ZZ(ld['p']),ld['ord_cond']) for ld in local_data]) Dfac = Factorization([(ZZ(ld['p']),ld['ord_disc']) for ld in local_data], unit=ZZ(self.signD)) data['minq_D'] = minqD = self.min_quad_twist['disc'] minq_label = self.min_quad_twist['label'] data['minq_label'] = db_ec().find_one({'label':minq_label}, ['lmfdb_label'])['lmfdb_label'] data['minq_info'] = '(itself)' if minqD==1 else '(by %s)' % minqD try: data['degree'] = self.degree except AttributeError: data['degree'] =0 # invalid, but will be displayed nicely mw['heights'] = self.heights if self.number == 1: data['an'] = self.anlist data['ap'] = self.aplist else: r = db_ec().find_one({'lmfdb_iso':self.lmfdb_iso, 'number':1}, ['anlist','aplist']) data['an'] = r['anlist'] data['ap'] = r['aplist'] # otherwise fall back to computing it from the curve except AttributeError: print("Falling back to constructing E") self.E = EllipticCurve(data['ainvs']) data['equation'] = web_latex(self.E) data['disc'] = D = self.E.discriminant() Nfac = N.factor() Dfac = D.factor() bad_primes = [p for p,e in Nfac] try: data['degree'] = self.degree except AttributeError: try: data['degree'] = self.E.modular_degree() except RuntimeError: data['degree'] = 0 # invalid, but will be displayed nicely minq, minqD = self.E.minimal_quadratic_twist() data['minq_D'] = minqD if minqD == 1: data['minq_label'] = self.lmfdb_label data['minq_info'] = '(itself)' else: # This relies on the minimal twist being in the # database, which is true when the database only # contains the Cremona database. It would be a good # idea if, when the database is extended, we ensured # that for any curve included, all twists of smaller # conductor are also included. minq_ainvs = [str(c) for c in minq.ainvs()] data['minq_label'] = db_ec().find_one({'jinv':str(self.E.j_invariant()), 'ainvs': minq_ainvs},['lmfdb_label'])['lmfdb_label'] data['minq_info'] = '(by %s)' % minqD if self.gens: self.generators = [self.E(g) for g in parse_points(self.gens)] mw['heights'] = [P.height() for P in self.generators] data['an'] = self.E.anlist(20,python_ints=True) data['ap'] = self.E.aplist(100,python_ints=True) self.local_data = local_data = [] for p in bad_primes: ld = self.E.local_data(p, algorithm="generic") local_data_p = {} local_data_p['p'] = p local_data_p['cp'] = ld.tamagawa_number() local_data_p['kod'] = web_latex(ld.kodaira_symbol()).replace('$', '') local_data_p['red'] = ld.bad_reduction_type() rootno = -ld.bad_reduction_type() if rootno==0: rootno = self.E.root_number(p) local_data_p['rootno'] = rootno local_data_p['ord_cond'] = ld.conductor_valuation() local_data_p['ord_disc'] = ld.discriminant_valuation() local_data_p['ord_den_j'] = max(0,-self.E.j_invariant().valuation(p)) local_data.append(local_data_p) # If we got the data from the database, the root numbers may # not have been stored there, so we have to compute them. If # there are additive primes this means constructing the curve. for ld in self.local_data: if not 'rootno' in ld: rootno = -ld['red'] if rootno==0: try: E = self.E except AttributeError: self.E = E = EllipticCurve(data['ainvs']) rootno = E.root_number(ld['p']) ld['rootno'] = rootno minq_N, minq_iso, minq_number = split_lmfdb_label(data['minq_label']) data['disc_factor'] = latex(Dfac) data['cond_factor'] =latex(Nfac) data['disc_latex'] = web_latex(D) data['cond_latex'] = web_latex(N) data['CMD'] = self.cm data['CM'] = "no" data['EndE'] = "\(\Z\)" if self.cm: 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'] data['ST'] = st_link_by_name(1,2,'N(U(1))') else: data['ST'] = st_link_by_name(1,2,'SU(2)') data['p_adic_primes'] = [p for i,p in enumerate(prime_range(5, 100)) if (N*data['ap'][i]) %p !=0] 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'])] if self.twoadic_gens: from sage.matrix.all import Matrix data['twoadic_gen_matrices'] = ','.join([latex(Matrix(2,2,M)) for M in self.twoadic_gens]) data['twoadic_rouse_url'] = ROUSE_URL_PREFIX + self.twoadic_label + ".html" # Leading term of L-function & BSD data bsd = self.bsd = {} r = self.rank if r >= 2: bsd['lder_name'] = "L^{(%s)}(E,1)/%s!" % (r,r) elif r: 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 tamagawa_numbers = [ZZ(ld['cp']) for ld in local_data] cp_fac = [cp.factor() for cp in tamagawa_numbers] cp_fac = [latex(cp) if len(cp)<2 else '('+latex(cp)+')' for cp in cp_fac] bsd['tamagawa_factors'] = r'\cdot'.join(cp_fac) bsd['tamagawa_product'] = prod(tamagawa_numbers) cond, iso, num = split_lmfdb_label(self.lmfdb_label) data['newform'] = web_latex(PowerSeriesRing(QQ, 'q')(data['an'], 20, check=True)) data['newform_label'] = self.newform_label = newform_label(cond,2,1,iso) self.newform_link = url_for("emf.render_elliptic_modular_forms", level=cond, weight=2, character=1, label=iso) self.newform_exists_in_db = is_newform_in_db(self.newform_label) self._code = None self.friends = [ ('Isogeny class ' + self.lmfdb_iso, url_for(".by_double_iso_label", conductor=N, iso_label=iso)), ('Minimal quadratic twist %s %s' % (data['minq_info'], data['minq_label']), url_for(".by_triple_label", conductor=minq_N, iso_label=minq_iso, number=minq_number)), ('All twists ', url_for(".rational_elliptic_curves", jinv=self.jinv)), ('L-function', url_for("l_functions.l_function_ec_page", label=self.lmfdb_label))] if not self.cm: if N<=300: self.friends += [('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=self.lmfdb_iso))] if N<=50: self.friends += [('Symmetric cube L-function', url_for("l_functions.l_function_ec_sym_page", power='3', label=self.lmfdb_iso))] if self.newform_exists_in_db: self.friends += [('Modular form ' + self.newform_label, self.newform_link)] self.downloads = [('Download coefficients of q-expansion', url_for(".download_EC_qexp", label=self.lmfdb_label, limit=1000)), ('Download all stored data', url_for(".download_EC_all", label=self.lmfdb_label)), ('Download Magma code', url_for(".ec_code_download", conductor=cond, iso=iso, number=num, label=self.lmfdb_label, download_type='magma')), ('Download Sage code', url_for(".ec_code_download", conductor=cond, iso=iso, number=num, label=self.lmfdb_label, download_type='sage')), ('Download GP code', url_for(".ec_code_download", conductor=cond, iso=iso, number=num, label=self.lmfdb_label, download_type='gp')) ] try: self.plot = encode_plot(self.E.plot()) except AttributeError: self.plot = encode_plot(EllipticCurve(data['ainvs']).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']) ] self.title = "Elliptic Curve %s (Cremona label %s)" % (self.lmfdb_label, self.label) self.bread = [('Elliptic Curves', url_for("ecnf.index")), ('$\Q$', url_for(".rational_elliptic_curves")), ('%s' % N, url_for(".by_conductor", conductor=N)), ('%s' % iso, url_for(".by_double_iso_label", conductor=N, iso_label=iso)), ('%s' % num,' ')] def code(self): if self._code == None: self.make_code_snippets() return self._code def make_code_snippets(self): # read in code.yaml from current directory: _curdir = os.path.dirname(os.path.abspath(__file__)) self._code = yaml.load(open(os.path.join(_curdir, "code.yaml"))) # Fill in placeholders for this specific curve: for lang in ['sage', 'pari', 'magma']: self._code['curve'][lang] = self._code['curve'][lang] % (self.data['ainvs'],self.label) return for k in self._code: if k != 'prompt': for lang in self._code[k]: self._code[k][lang] = self._code[k][lang].split("\n") # remove final empty line if len(self._code[k][lang][-1])==0: self._code[k][lang] = self._code[k][lang][:-1]
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, }