def set_info_for_web_newform(level=None, weight=None, character=None, label=None, **kwds): r""" Set the info for on modular form. """ info = to_dict(kwds) info['level'] = level info['weight'] = weight info['character'] = character info['label'] = label if level is None or weight is None or character is None or label is None: s = "In set info for one form but do not have enough args!" s += "level={0},weight={1},character={2},label={3}".format(level, weight, character, label) emf_logger.critical(s) emf_logger.debug("In set_info_for_one_mf: info={0}".format(info)) prec = my_get(info, 'prec', default_prec, int) bprec = my_get(info, 'bprec', default_display_bprec, int) emf_logger.debug("PREC: {0}".format(prec)) emf_logger.debug("BITPREC: {0}".format(bprec)) try: WNF = WebNewForm_cached(level=level, weight=weight, character=character, label=label) emf_logger.critical("defined webnewform for rendering!") # if info.has_key('download') and info.has_key('tempfile'): # WNF._save_to_file(info['tempfile']) # info['filename']=str(weight)+'-'+str(level)+'-'+str(character)+'-'+label+'.sobj' # return info except IndexError as e: WNF = None info['error'] = e.message url1 = url_for("emf.render_elliptic_modular_forms") url2 = url_for("emf.render_elliptic_modular_forms", level=level) url3 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight) url4 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character) bread = [(EMF_TOP, url1)] bread.append(("of level %s" % level, url2)) bread.append(("weight %s" % weight, url3)) if int(character) == 0: bread.append(("trivial character", url4)) else: bread.append(("\( %s \)" % (WNF.character.latex_name), url4)) info['bread'] = bread properties2 = list() friends = list() space_url = url_for('emf.render_elliptic_modular_forms',level=level, weight=weight, character=character) friends.append(('\( S_{%s}(%s, %s)\)'%(WNF.weight, WNF.level, WNF.character.latex_name), space_url)) if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url: friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty, WNF.base_ring.lmfdb_url)) if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_label: friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_url)) friends = uniq(friends) friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)", WNF.character.url())) if WNF.dimension==0: info['error'] = "This space is empty!" # emf_logger.debug("WNF={0}".format(WNF)) #info['name'] = name info['title'] = 'Modular Form ' + WNF.hecke_orbit_label if 'error' in info: return info # info['name']=WNF._name ## Until we have figured out how to do the embeddings correctly we don't display the Satake ## parameters for non-trivial characters.... ## Example to illustrate the different cases ## base = CyclotomicField(n) -- of degree phi(n) ## coefficient_field = NumberField( p(x)) for some p in base['x'] of degree m ## we would then have cdeg = m*phi(n) and bdeg = phi(n) ## and rdeg = m ## Unfortunately, for e.g. base = coefficient_field = CyclotomicField(6) ## we get coefficient_field.relative_degree() == 2 although it should be 1 cdeg = WNF.coefficient_field.absolute_degree() bdeg = WNF.base_ring.absolute_degree() if cdeg == 1: rdeg = 1 else: ## just setting rdeg = WNF.coefficient_field.relative_degree() does not give correct result... ## rdeg = QQ(cdeg)/QQ(bdeg) cf_is_QQ = (cdeg == 1) br_is_QQ = (bdeg == 1) if cf_is_QQ: info['satake'] = WNF.satake info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ') info['qexp_display'] = url_for(".get_qexp_latex", level=level, weight=weight, character=character, label=label) info['max_cn_qexp'] = WNF.q_expansion.prec() ## All combinations should be tested... ## 13/4/4/a -> base ring = coefficient_field = QQ(zeta_6) ## 13/3/8/a -> base_ring = QQ(zeta_4), coefficient_field has poly x^2+(2\zeta_4+2x-3\zeta_$ over base_ring ## 13/4/3/a -> base_ring = coefficient_field = QQ(zeta_3) ## 13/4/1/a -> all rational ## 13/6/1/a/ -> base_ring = QQ, coefficient_field = Q(sqrt(17)) ## These are variables which needs to be set properly below info['polvars'] = {'base_ring':'x','coefficient_field':'\\alpha'} if not cf_is_QQ: if rdeg>1: # not WNF.coefficient_field == WNF.base_ring: ## Here WNF.base_ring should be some cyclotomic field and we have an extension over this. p1 = WNF.coefficient_field.relative_polynomial() c_pol_ltx = web_latex_poly(p1, '\\alpha') # make the variable \alpha c_pol_ltx_x = web_latex_poly(p1, 'x') zeta = p1.base_ring().gens()[0] # p2 = zeta.minpoly() #this is not used anymore # b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore z1 = zeta.multiplicative_order() info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'),c_pol_ltx_x, z1] if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url: info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label] if z1==4: info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format(c_pol_ltx) info['polvars']['base_ring']='i' elif z1<=2: info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(c_pol_ltx) else: info['polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\) '%(c_pol_ltx, z1,z1) info['polvars']['base_ring']='\zeta_{{ {0} }}'.format(z1) if z1==3: info['polynomial_st'] += 'is a primitive cube root of unity.' else: info['polynomial_st'] += 'is a primitive {0}-th root of unity.'.format(z1) elif not br_is_QQ: ## Now we have base and coefficient field being equal, meaning that since the coefficient field is not QQ it is some cyclotomic field ## generated by some \zeta_n p1 = WNF.coefficient_field.absolute_polynomial() z1 = WNF.coefficient_field.gens()[0].multiplicative_order() c_pol_ltx = web_latex_poly(p1, '\\zeta_{{{0}}}'.format(z1)) c_pol_ltx_x = web_latex_poly(p1, 'x') info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'), c_pol_ltx_x] if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url: info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label] if z1==4: info['polynomial_st'] = '<div class="where">where \(\zeta_4=e^{{\\frac{{\\pi i}}{{ 2 }} }}=i \).</div>'.format(c_pol_ltx) info['polvars']['coefficient_field']='i' elif z1<=2: info['polynomial_st'] = '' else: info['polynomial_st'] = '<div class="where">where \(\zeta_{{{0}}}=e^{{\\frac{{2\\pi i}}{{ {0} }} }}\) '.format(z1) info['polvars']['coefficient_field']='\zeta_{{{0}}}'.format(z1) if z1==3: info['polynomial_st'] += 'is a primitive cube root of unity.</div>' else: info['polynomial_st'] += 'is a primitive {0}-th root of unity.</div>'.format(z1) else: info['polynomial_st'] = '' info['degree'] = int(cdeg) if cdeg==1: info['is_rational'] = 1 info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty ] else: info['is_rational'] = 0 # info['q_exp_embeddings'] = WNF.print_q_expansion_embeddings() # if(int(info['degree'])>1 and WNF.dimension()>1): # s = 'One can embed it into \( \mathbb{C} \) as:' # bprec = 26 # print s # info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision']) # elif(int(info['degree'])>1): # s = 'There are '+str(info['degree'])+' embeddings into \( \mathbb{C} \):' # bprec = 26 # print s # info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision']) # else: # info['embeddings'] = '' emf_logger.debug("PREC2: {0}".format(prec)) info['embeddings'] = WNF._embeddings['values'] #q_expansion_embeddings(prec, bprec,format='latex') info['embeddings_len'] = len(info['embeddings']) properties2 = [] if (ZZ(level)).is_squarefree(): info['twist_info'] = WNF.twist_info if isinstance(info['twist_info'], list) and len(info['twist_info'])>0: info['is_minimal'] = info['twist_info'][0] if(info['twist_info'][0]): s = 'Is minimal<br>' else: s = 'Is a twist of lower level<br>' properties2 = [('Twist info', s)] else: info['twist_info'] = 'Twist info currently not available.' properties2 = [('Twist info', 'not available')] args = list() for x in range(5, 200, 10): args.append({'digits': x}) alev = None CM = WNF._cm_values if CM is not None: if CM.has_key('tau') and len(CM['tau']) != 0: info['CM_values'] = CM info['is_cm'] = WNF.is_cm if WNF.is_cm == 1: info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc) info['cm_disc'] = WNF.cm_disc info['cm_field_knowl'] = nf_display_knowl(info['cm_field'], getDBConnection(), field_pretty(info['cm_field'])) info['cm_field_url'] = url_for("number_fields.by_label", label=info["cm_field"]) if WNF.is_cm is None or WNF.is_cm==-1: s = '- Unknown (insufficient data)<br>' elif WNF.is_cm == 1: s = 'Is a CM-form<br>' else: s = 'Is not a CM-form<br>' properties2.append(('CM info', s)) alev = WNF.atkin_lehner_eigenvalues() info['atkinlehner'] = None if isinstance(alev,dict) and len(alev.keys())>0 and level != 1: s1 = " Atkin-Lehner eigenvalues " s2 = "" for Q in alev.keys(): s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q]) properties2.append((s1, s2)) emf_logger.debug("properties={0}".format(properties2)) # alev = WNF.atkin_lehner_eigenvalues_for_all_cusps() # if isinstance(alev,dict) and len(alev.keys())>0: # emf_logger.debug("alev={0}".format(alev)) # info['atkinlehner'] = list() # for Q in alev.keys(): # s = "\(" + latex(c) + "\)" # Q = alev[c][0] # ev = alev[c][1] # info['atkinlehner'].append([Q, c, ev]) if(level == 1): poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6','') if poly != '': d,monom,coeffs = poly emf_logger.critical("poly={0}".format(poly)) info['explicit_formulas'] = '\(' for i in range(len(coeffs)): c = QQ(coeffs[i]) s = "" if d>1 and i >0 and c>0: s="+" if c<0: s="-" if c.denominator()>1: cc = "\\frac{{ {0} }}{{ {1} }}".format(abs(c.numerator()),c.denominator()) else: cc = str(abs(c)) s += "{0} \cdot ".format(cc) a = monom[i][0]; b = monom[i][1] if a == 0 and b != 0: s+="E_6^{{ {0} }}".format(b) elif b ==0 and a != 0: s+="E_4^{{ {0} }}".format(a) else: s+="E_4^{{ {0} }}E_6^{{ {1} }}".format(a,b) info['explicit_formulas'] += s info['explicit_formulas'] += " \)" cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + \ '&label=' + str(label) if len(WNF.parent.hecke_orbits) > 1: for label_other in WNF.parent.hecke_orbits.keys(): if(label_other != label): s = 'Modular form ' if character: s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label_other) else: s = s + str(level) + '.' + str(weight) + str(label_other) url = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label_other) friends.append((s, url)) s = 'L-Function ' if character: s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label) else: s = s + str(level) + '.' + str(weight) + str(label) # url = # "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s" # %(level,weight,character,label,0) url = '/L' + url_for( 'emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label) if WNF.coefficient_field_degree > 1: for h in range(WNF.coefficient_field_degree): s0 = s + ".{0}".format(h) url0 = url + "{0}/".format(h) friends.append((s0, url0)) else: friends.append((s, url)) # if there is an elliptic curve over Q associated to self we also list that if WNF.weight == 2 and WNF.coefficient_field_degree == 1: llabel = str(level) + '.' + label s = 'Elliptic curve isogeny class ' + llabel url = '/EllipticCurve/Q/' + llabel friends.append((s, url)) info['properties2'] = properties2 info['friends'] = friends info['max_cn'] = WNF.max_cn() return info
def set_info_for_web_newform(level=None, weight=None, character=None, label=None, **kwds): r""" Set the info for on modular form. """ info = to_dict(kwds) info['level'] = level info['weight'] = weight info['character'] = character info['label'] = label if level is None or weight is None or character is None or label is None: s = "In set info for one form but do not have enough args!" s += "level={0},weight={1},character={2},label={3}".format( level, weight, character, label) emf_logger.critical(s) emf_logger.debug("In set_info_for_one_mf: info={0}".format(info)) prec = my_get(info, 'prec', default_prec, int) bprec = my_get(info, 'bprec', default_display_bprec, int) emf_logger.debug("PREC: {0}".format(prec)) emf_logger.debug("BITPREC: {0}".format(bprec)) try: WNF = WebNewForm_cached(level=level, weight=weight, character=character, label=label, prec=prec) emf_logger.debug("defined webnewform for rendering!") except IndexError as e: WNF = None info['error'] = e.message url1 = url_for("emf.render_elliptic_modular_forms") url2 = url_for("emf.render_elliptic_modular_forms", level=level) url3 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight) url4 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character) bread = [(EMF_TOP, url1)] bread.append(("Level %s" % level, url2)) bread.append(("Weight %s" % weight, url3)) if int(character) == 0: bread.append(("Trivial Character", url4)) else: bread.append(("Character \( %s \)" % (WNF.character.latex_name), url4)) bread.append( ("Newform %d.%d.%d.%s" % (level, weight, int(character), label), '')) info['bread'] = bread properties2 = list() friends = list() space_url = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character=character) friends.append( ('\( S_{%s}(%s, %s)\)' % (WNF.weight, WNF.level, WNF.character.latex_name), space_url)) if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url: friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty, WNF.base_ring.lmfdb_url)) if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_label: friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_url)) friends = uniq(friends) friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)", WNF.character.url())) if WNF.dimension == 0: info['error'] = "This space is empty!" info['title'] = 'Newform ' + WNF.hecke_orbit_label info['learnmore'] = [('History of Modular forms', url_for('holomorphic_mf_history'))] if 'error' in info: return info ## Until we have figured out how to do the embeddings correctly we don't display the Satake ## parameters for non-trivial characters.... ## Example to illustrate the different cases ## base = CyclotomicField(n) -- of degree phi(n) ## coefficient_field = NumberField( p(x)) for some p in base['x'] of degree m ## we would then have cdeg = m*phi(n) and bdeg = phi(n) ## and rdeg = m ## Unfortunately, for e.g. base = coefficient_field = CyclotomicField(6) ## we get coefficient_field.relative_degree() == 2 although it should be 1 cdeg = WNF.coefficient_field.absolute_degree() bdeg = WNF.base_ring.absolute_degree() if cdeg == 1: rdeg = 1 else: ## just setting rdeg = WNF.coefficient_field.relative_degree() does not give correct result... ## rdeg = QQ(cdeg) / QQ(bdeg) cf_is_QQ = (cdeg == 1) br_is_QQ = (bdeg == 1) if cf_is_QQ: info['satake'] = WNF.satake if WNF.complexity_of_first_nonvanishing_coefficients( ) > default_max_height: info['qexp'] = "" info['qexp_display'] = '' info['hide_qexp'] = True n, c = WNF.first_nonvanishing_coefficient() info['trace_nv'] = latex(c.trace()) info['norm_nv'] = '\\approx ' + latex(c.norm().n()) info['index_nv'] = n else: info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ') info['qexp_display'] = url_for(".get_qexp_latex", level=level, weight=weight, character=character, label=label) info["hide_qexp"] = False info['max_cn_qexp'] = WNF.q_expansion.prec() ## All combinations should be tested... ## 13/4/4/a -> base ring = coefficient_field = QQ(zeta_6) ## 13/3/8/a -> base_ring = QQ(zeta_4), coefficient_field has poly x^2+(2\zeta_4+2x-3\zeta_$ over base_ring ## 13/4/3/a -> base_ring = coefficient_field = QQ(zeta_3) ## 13/4/1/a -> all rational ## 13/6/1/a/ -> base_ring = QQ, coefficient_field = Q(sqrt(17)) ## These are variables which needs to be set properly below info['polvars'] = {'base_ring': 'x', 'coefficient_field': '\\alpha'} if not cf_is_QQ: if rdeg > 1: # not WNF.coefficient_field == WNF.base_ring: ## Here WNF.base_ring should be some cyclotomic field and we have an extension over this. p1 = WNF.coefficient_field.relative_polynomial() c_pol_ltx = web_latex_poly(p1, '\\alpha') # make the variable \alpha c_pol_ltx_x = web_latex_poly(p1, 'x') zeta = p1.base_ring().gens()[0] # p2 = zeta.minpoly() #this is not used anymore # b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore z1 = zeta.multiplicative_order() info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'), c_pol_ltx_x, z1 ] if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url: info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label ] if z1 == 4: info[ 'polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format( c_pol_ltx) info['polvars']['base_ring'] = 'i' elif z1 <= 2: info[ 'polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format( c_pol_ltx) else: info[ 'polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\) ' % ( c_pol_ltx, z1, z1) info['polvars']['base_ring'] = '\zeta_{{ {0} }}'.format(z1) if z1 == 3: info[ 'polynomial_st'] += 'is a primitive cube root of unity.' else: info[ 'polynomial_st'] += 'is a primitive {0}-th root of unity.'.format( z1) elif not br_is_QQ: ## Now we have base and coefficient field being equal, meaning that since the coefficient field is not QQ it is some cyclotomic field ## generated by some \zeta_n p1 = WNF.coefficient_field.absolute_polynomial() z1 = WNF.coefficient_field.gens()[0].multiplicative_order() c_pol_ltx = web_latex_poly(p1, '\\zeta_{{{0}}}'.format(z1)) c_pol_ltx_x = web_latex_poly(p1, 'x') info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'), c_pol_ltx_x ] if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url: info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label ] if z1 == 4: info[ 'polynomial_st'] = '<div class="where">where \(\zeta_4=e^{{\\frac{{\\pi i}}{{ 2 }} }}=i \).</div>'.format( c_pol_ltx) info['polvars']['coefficient_field'] = 'i' elif z1 <= 2: info['polynomial_st'] = '' else: info[ 'polynomial_st'] = '<div class="where">where \(\zeta_{{{0}}}=e^{{\\frac{{2\\pi i}}{{ {0} }} }}\) '.format( z1) info['polvars']['coefficient_field'] = '\zeta_{{{0}}}'.format( z1) if z1 == 3: info[ 'polynomial_st'] += 'is a primitive cube root of unity.</div>' else: info[ 'polynomial_st'] += 'is a primitive {0}-th root of unity.</div>'.format( z1) else: info['polynomial_st'] = '' if info["hide_qexp"]: info['polynomial_st'] = '' info['degree'] = int(cdeg) if cdeg == 1: info['is_rational'] = 1 info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty ] else: info['is_rational'] = 0 emf_logger.debug("PREC2: {0}".format(prec)) info['embeddings'] = WNF._embeddings[ 'values'] #q_expansion_embeddings(prec, bprec,format='latex') info['embeddings_len'] = len(info['embeddings']) properties2 = [('Level', str(level)), ('Weight', str(weight)), ('Character', '$' + WNF.character.latex_name + '$'), ('Label', WNF.hecke_orbit_label), ('Dimension of Galois orbit', str(WNF.dimension))] if (ZZ(level)).is_squarefree(): info['twist_info'] = WNF.twist_info if isinstance(info['twist_info'], list) and len(info['twist_info']) > 0: info['is_minimal'] = info['twist_info'][0] if (info['twist_info'][0]): s = 'Is minimal<br>' else: s = 'Is a twist of lower level<br>' properties2 += [('Twist info', s)] else: info['twist_info'] = 'Twist info currently not available.' properties2 += [('Twist info', 'not available')] args = list() for x in range(5, 200, 10): args.append({'digits': x}) alev = None CM = WNF._cm_values if CM is not None: if CM.has_key('tau') and len(CM['tau']) != 0: info['CM_values'] = CM info['is_cm'] = WNF.is_cm if WNF.is_cm == 1: info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc) info['cm_disc'] = WNF.cm_disc info['cm_field_knowl'] = nf_display_knowl( info['cm_field'], getDBConnection(), field_pretty(info['cm_field'])) info['cm_field_url'] = url_for("number_fields.by_label", label=info["cm_field"]) if WNF.is_cm is None or WNF.is_cm == -1: s = '- Unknown (insufficient data)<br>' elif WNF.is_cm == 1: s = 'Yes<br>' else: s = 'No<br>' properties2.append(('CM', s)) alev = WNF.atkin_lehner_eigenvalues() info['atkinlehner'] = None if isinstance(alev, dict) and len(alev.keys()) > 0 and level != 1: s1 = " Atkin-Lehner eigenvalues " s2 = "" for Q in alev.keys(): s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q]) properties2.append((s1, s2)) emf_logger.debug("properties={0}".format(properties2)) # alev = WNF.atkin_lehner_eigenvalues_for_all_cusps() # if isinstance(alev,dict) and len(alev.keys())>0: # emf_logger.debug("alev={0}".format(alev)) # info['atkinlehner'] = list() # for Q in alev.keys(): # s = "\(" + latex(c) + "\)" # Q = alev[c][0] # ev = alev[c][1] # info['atkinlehner'].append([Q, c, ev]) if (level == 1): poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6', '') if poly != '': d, monom, coeffs = poly emf_logger.critical("poly={0}".format(poly)) info['explicit_formulas'] = '\(' for i in range(len(coeffs)): c = QQ(coeffs[i]) s = "" if d > 1 and i > 0 and c > 0: s = "+" if c < 0: s = "-" if c.denominator() > 1: cc = "\\frac{{ {0} }}{{ {1} }}".format( abs(c.numerator()), c.denominator()) else: cc = str(abs(c)) s += "{0} \cdot ".format(cc) a = monom[i][0] b = monom[i][1] if a == 0 and b != 0: s += "E_6^{{ {0} }}".format(b) elif b == 0 and a != 0: s += "E_4^{{ {0} }}".format(a) else: s += "E_4^{{ {0} }}E_6^{{ {1} }}".format(a, b) info['explicit_formulas'] += s info['explicit_formulas'] += " \)" cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + \ '&label=' + str(label) if len(WNF.parent.hecke_orbits) > 1: for label_other in WNF.parent.hecke_orbits.keys(): if (label_other != label): s = 'Modular form ' if character: s += newform_label(level, weight, character, label_other) else: s += newform_label(level, weight, 1, label_other) url = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label_other) friends.append((s, url)) s = 'L-Function ' if character: s += newform_label(level, weight, character, label) else: s += newform_label(level, weight, 1, label) # url = # "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s" # %(level,weight,character,label,0) url = '/L' + url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label) if WNF.coefficient_field_degree > 1: for h in range(WNF.coefficient_field_degree): s0 = s + ".{0}".format(h) url0 = url + "{0}/".format(h) friends.append((s0, url0)) else: friends.append((s, url)) # if there is an elliptic curve over Q associated to self we also list that if WNF.weight == 2 and WNF.coefficient_field_degree == 1: llabel = str(level) + '.' + label s = 'Elliptic curve isogeny class ' + llabel url = '/EllipticCurve/Q/' + llabel friends.append((s, url)) info['properties2'] = properties2 info['friends'] = friends info['max_cn'] = WNF.max_available_prec() return info
def make_class(self): # Create a list of the curves in the class from the database self.db_curves = [ c for c in db_ec().find({ 'field_label': self.field_label, 'conductor_label': self.conductor_label, 'iso_label': self.iso_label }).sort('number') ] size = len(self.db_curves) # Rank or bounds try: self.rk = web_latex(self.db_curves[0]['rank']) except KeyError: self.rk = "?" try: self.rk_bnds = "%s...%s" % tuple(self.db_curves[0]['rank_bounds']) except KeyError: self.rank_bounds = [0, sage.rings.infinity.Infinity] self.rk_bnds = "not recorded" # Extract the isogeny degree matrix from the database if possible, else create it if hasattr(self, 'isogeny_matrix'): from sage.matrix.all import Matrix self.isogeny_matrix = Matrix(self.isogeny_matrix) else: self.isogeny_matrix = make_iso_matrix(self.db_curves) # Create isogeny graph: self.graph = make_graph(self.isogeny_matrix) P = self.graph.plot(edge_labels=True) self.graph_img = encode_plot(P) self.graph_link = '<img src="%s" width="200" height="150"/>' % self.graph_img self.isogeny_matrix_str = latex(matrix(self.isogeny_matrix)) self.field = field_pretty(self.field_label) self.field_knowl = nf_display_knowl(self.field_label, lmfdb.base.getDBConnection(), self.field) def curve_url(c): return url_for(".show_ecnf", nf=c['field_label'], conductor_label=c['conductor_label'], class_label=c['iso_label'], number=c['number']) self.curves = [[ c['short_label'], curve_url(c), web_ainvs(self.field_label, c['ainvs']) ] for c in self.db_curves] self.urls = {} self.urls['class'] = url_for(".show_ecnf_isoclass", nf=self.field_label, conductor_label=self.conductor_label, class_label=self.iso_label) self.urls['conductor'] = url_for(".show_ecnf_conductor", nf=self.field_label, conductor_label=self.conductor_label) self.urls['field'] = url_for('.show_ecnf1', nf=self.field_label) sig = self.signature real_quadratic = sig == [2, 0] totally_real = sig[1] == 0 imag_quadratic = sig == [0, 1] if totally_real: self.hmf_label = "-".join( [self.field_label, self.conductor_label, self.iso_label]) self.urls['hmf'] = url_for('hmf.render_hmf_webpage', field_label=self.field_label, label=self.hmf_label) self.urls['Lfunction'] = url_for("l_functions.l_function_hmf_page", field=self.field_label, label=self.hmf_label, character='0', number='0') if imag_quadratic: self.bmf_label = "-".join( [self.field_label, self.conductor_label, self.iso_label]) self.friends = [] if totally_real: self.friends += [('Hilbert Modular Form ' + self.hmf_label, self.urls['hmf'])] self.friends += [('L-function', self.urls['Lfunction'])] if imag_quadratic: self.friends += [ ('Bianchi Modular Form %s not available' % self.bmf_label, '') ] self.properties = [('Base field', self.field), ('Label', self.class_label), (None, self.graph_link), ('Conductor', '%s' % self.conductor_label)] if self.rk != '?': self.properties += [('Rank', '%s' % self.rk)] else: if self.rk_bnds == 'not recorded': self.properties += [('Rank', '%s' % self.rk_bnds)] else: self.properties += [('Rank bounds', '%s' % self.rk_bnds)] self.bread = [('Elliptic Curves ', url_for(".index")), (self.field_label, self.urls['field']), (self.conductor_label, self.urls['conductor']), ('isogeny class %s' % self.short_label, self.urls['class'])]
def make_class(self): # Create a list of the curves in the class from the database self.db_curves = [c for c in db_ec().find( {'field_label': self.field_label, 'conductor_label': self.conductor_label, 'iso_label': self.iso_label}).sort('number')] size = len(self.db_curves) # Extract the isogeny degree matrix from the database if possible, else create it if hasattr(self, 'isogeny_matrix'): from sage.matrix.all import Matrix self.isogeny_matrix = Matrix(self.isogeny_matrix) else: self.isogeny_matrix = make_iso_matrix(self.db_curves) # Create isogeny graph: self.graph = make_graph(self.isogeny_matrix) P = self.graph.plot(edge_labels=True) self.graph_img = encode_plot(P) self.graph_link = '<img src="%s" width="200" height="150"/>' % self.graph_img self.isogeny_matrix_str = latex(matrix(self.isogeny_matrix)) self.field = field_pretty(self.field_label) self.field_knowl = nf_display_knowl(self.field_label, lmfdb.base.getDBConnection(), self.field) def curve_url(c): return url_for(".show_ecnf", nf=c['field_label'], conductor_label=c['conductor_label'], class_label=c['iso_label'], number=c['number']) self.curves = [[c['short_label'], curve_url(c), web_ainvs(self.field_label,c['ainvs'])] for c in self.db_curves] self.urls = {} self.urls['class'] = url_for(".show_ecnf_isoclass", nf=self.field_label, conductor_label=self.conductor_label, class_label=self.iso_label) self.urls['conductor'] = url_for(".show_ecnf_conductor", nf=self.field_label, conductor_label=self.conductor_label) self.urls['field'] = url_for('.show_ecnf1', nf=self.field_label) real_quadratic = self.signature == [2,0] imag_quadratic = self.signature == [0,1] if real_quadratic: self.hmf_label = "-".join([self.field_label, self.conductor_label, self.iso_label]) self.urls['hmf'] = url_for('hmf.render_hmf_webpage', field_label=self.field_label, label=self.hmf_label) if imag_quadratic: self.bmf_label = "-".join([self.field_label, self.conductor_label, self.iso_label]) self.friends = [] if real_quadratic: self.friends += [('Hilbert Modular Form ' + self.hmf_label, self.urls['hmf'])] if imag_quadratic: self.friends += [('Bianchi Modular Form %s not yet available' % self.bmf_label, '')] self.properties = [('Base field', self.field), ('Label', self.class_label), (None, self.graph_link), ('Conductor', '%s' % self.conductor_label) ] self.bread = [('Elliptic Curves ', url_for(".index")), (self.field_label, self.urls['field']), (self.conductor_label, self.urls['conductor']), ('isogeny class %s' % self.short_label, self.urls['class'])]
def set_info_for_web_newform(level=None, weight=None, character=None, label=None, **kwds): r""" Set the info for on modular form. """ info = to_dict(kwds) info['level'] = level info['weight'] = weight info['character'] = character info['label'] = label if level is None or weight is None or character is None or label is None: s = "In set info for one form but do not have enough args!" s += "level={0},weight={1},character={2},label={3}".format(level, weight, character, label) emf_logger.critical(s) emf_logger.debug("In set_info_for_one_mf: info={0}".format(info)) prec = my_get(info, 'prec', default_prec, int) bprec = my_get(info, 'bprec', default_display_bprec, int) emf_logger.debug("PREC: {0}".format(prec)) emf_logger.debug("BITPREC: {0}".format(bprec)) try: WNF = WebNewForm_cached(level=level, weight=weight, character=character, label=label) emf_logger.critical("defined webnewform for rendering!") # if info.has_key('download') and info.has_key('tempfile'): # WNF._save_to_file(info['tempfile']) # info['filename']=str(weight)+'-'+str(level)+'-'+str(character)+'-'+label+'.sobj' # return info except IndexError as e: WNF = None info['error'] = e.message url1 = url_for("emf.render_elliptic_modular_forms") url2 = url_for("emf.render_elliptic_modular_forms", level=level) url3 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight) url4 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character) bread = [(EMF_TOP, url1)] bread.append(("of level %s" % level, url2)) bread.append(("weight %s" % weight, url3)) if int(character) == 0: bread.append(("trivial character", url4)) else: bread.append(("\( %s \)" % (WNF.character.latex_name), url4)) info['bread'] = bread properties2 = list() friends = list() space_url = url_for('emf.render_elliptic_modular_forms',level=level, weight=weight, character=character) friends.append(('\( S_{%s}(%s, %s)\)'%(WNF.weight, WNF.level, WNF.character.latex_name), space_url)) if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url: friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty, WNF.base_ring.lmfdb_url)) if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_label: friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_url)) friends = uniq(friends) friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)", WNF.character.url())) if WNF.dimension==0: info['error'] = "This space is empty!" # emf_logger.debug("WNF={0}".format(WNF)) #info['name'] = name info['title'] = 'Modular Form ' + WNF.hecke_orbit_label if 'error' in info: return info # info['name']=WNF._name ## Until we have figured out how to do the embeddings correctly we don't display the Satake ## parameters for non-trivial characters.... cdeg = WNF.coefficient_field.absolute_degree() bdeg = WNF.base_ring.absolute_degree() if cdeg == 1: rdeg = 1 else: rdeg = WNF.coefficient_field.relative_degree() cf_is_QQ = (cdeg == 1) br_is_QQ = (bdeg == 1) if cf_is_QQ: info['satake'] = WNF.satake info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ') info['qexp_display'] = url_for(".get_qexp_latex", level=level, weight=weight, character=character, label=label) info['max_cn_qexp'] = WNF.q_expansion.prec() if not cf_is_QQ: if not br_is_QQ and rdeg>1: # not WNF.coefficient_field == WNF.base_ring: p1 = WNF.coefficient_field.relative_polynomial() c_pol_ltx = web_latex_poly(p1, '\\alpha') # make the variable \alpha c_pol_ltx_x = web_latex_poly(p1, 'x') zeta = p1.base_ring().gens()[0] # p2 = zeta.minpoly() #this is not used anymore # b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore z1 = zeta.multiplicative_order() info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'),c_pol_ltx_x, z1] if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url: info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label] if z1==4: info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format(c_pol_ltx) elif z1<=2: info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(c_pol_ltx) else: info['polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\).'%(c_pol_ltx, z1,z1) else: p1 = WNF.coefficient_field.relative_polynomial() c_pol_ltx = web_latex_poly(p1, '\\alpha') c_pol_ltx_x = web_latex_poly(p1, 'x') z1 = p1.base_ring().gens()[0].multiplicative_order() info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'), c_pol_ltx_x, z1] if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url: info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label] if z1==4: info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).'.format(c_pol_ltx) elif z1<=2: info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(c_pol_ltx) else: info['polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\).'%(c_pol_ltx, z1,z1) else: info['polynomial_st'] = '' info['degree'] = int(cdeg) if cdeg==1: info['is_rational'] = 1 info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty ] else: info['is_rational'] = 0 # info['q_exp_embeddings'] = WNF.print_q_expansion_embeddings() # if(int(info['degree'])>1 and WNF.dimension()>1): # s = 'One can embed it into \( \mathbb{C} \) as:' # bprec = 26 # print s # info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision']) # elif(int(info['degree'])>1): # s = 'There are '+str(info['degree'])+' embeddings into \( \mathbb{C} \):' # bprec = 26 # print s # info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision']) # else: # info['embeddings'] = '' emf_logger.debug("PREC2: {0}".format(prec)) info['embeddings'] = WNF._embeddings['values'] #q_expansion_embeddings(prec, bprec,format='latex') info['embeddings_len'] = len(info['embeddings']) properties2 = [] if (ZZ(level)).is_squarefree(): info['twist_info'] = WNF.twist_info if isinstance(info['twist_info'], list) and len(info['twist_info'])>0: info['is_minimal'] = info['twist_info'][0] if(info['twist_info'][0]): s = 'Is minimal<br>' else: s = 'Is a twist of lower level<br>' properties2 = [('Twist info', s)] else: info['twist_info'] = 'Twist info currently not available.' properties2 = [('Twist info', 'not available')] args = list() for x in range(5, 200, 10): args.append({'digits': x}) alev = None CM = WNF._cm_values if CM is not None: if CM.has_key('tau') and len(CM['tau']) != 0: info['CM_values'] = CM info['is_cm'] = WNF.is_cm if WNF.is_cm: info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc) info['cm_disc'] = WNF.cm_disc info['cm_field_knowl'] = nf_display_knowl(info['cm_field'], getDBConnection(), field_pretty(info['cm_field'])) info['cm_field_url'] = url_for("number_fields.by_label", label=info["cm_field"]) if WNF.is_cm is None: s = '- Unknown (insufficient data)<br>' elif WNF.is_cm: s = 'Is a CM-form<br>' else: s = 'Is not a CM-form<br>' properties2.append(('CM info', s)) alev = WNF.atkin_lehner_eigenvalues() info['atkinlehner'] = None if isinstance(alev,dict) and len(alev.keys())>0 and level != 1: s1 = " Atkin-Lehner eigenvalues " s2 = "" for Q in alev.keys(): s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q]) properties2.append((s1, s2)) emf_logger.debug("properties={0}".format(properties2)) # alev = WNF.atkin_lehner_eigenvalues_for_all_cusps() # if isinstance(alev,dict) and len(alev.keys())>0: # emf_logger.debug("alev={0}".format(alev)) # info['atkinlehner'] = list() # for Q in alev.keys(): # s = "\(" + latex(c) + "\)" # Q = alev[c][0] # ev = alev[c][1] # info['atkinlehner'].append([Q, c, ev]) if(level == 1): poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6','') if poly != '': d,monom,coeffs = poly emf_logger.critical("poly={0}".format(poly)) info['explicit_formulas'] = '\(' for i in range(len(coeffs)): c = QQ(coeffs[i]) s = "" if d>1 and i >0 and c>0: s="+" if c<0: s="-" if c.denominator()>1: cc = "\\frac{{ {0} }}{{ {1} }}".format(abs(c.numerator()),c.denominator()) else: cc = str(abs(c)) s += "{0} \cdot ".format(cc) a = monom[i][0]; b = monom[i][1] if a == 0 and b != 0: s+="E_6^{{ {0} }}".format(b) elif b ==0 and a != 0: s+="E_4^{{ {0} }}".format(a) else: s+="E_4^{{ {0} }}E_6^{{ {1} }}".format(a,b) info['explicit_formulas'] += s info['explicit_formulas'] += " \)" cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + \ '&label=' + str(label) if len(WNF.parent.hecke_orbits) > 1: for label_other in WNF.parent.hecke_orbits.keys(): if(label_other != label): s = 'Modular form ' if character: s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label_other) else: s = s + str(level) + '.' + str(weight) + str(label_other) url = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label_other) friends.append((s, url)) s = 'L-Function ' if character: s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label) else: s = s + str(level) + '.' + str(weight) + str(label) # url = # "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s" # %(level,weight,character,label,0) url = '/L' + url_for( 'emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label) if WNF.coefficient_field_degree > 1: for h in range(WNF.coefficient_field_degree): s0 = s + ".{0}".format(h) url0 = url + "{0}/".format(h) friends.append((s0, url0)) else: friends.append((s, url)) # if there is an elliptic curve over Q associated to self we also list that if WNF.weight == 2 and WNF.coefficient_field_degree == 1: llabel = str(level) + '.' + label s = 'Elliptic curve isogeny class ' + llabel url = '/EllipticCurve/Q/' + llabel friends.append((s, url)) info['properties2'] = properties2 info['friends'] = friends info['max_cn'] = WNF.max_cn() return info
def make_class(self): # Create a list of the curves in the class from the database self.db_curves = [c for c in db_ec().find( {'field_label': self.field_label, 'conductor_label': self.conductor_label, 'iso_label': self.iso_label}).sort('number')] size = len(self.db_curves) # Rank or bounds try: self.rk = web_latex(self.db_curves[0]['rank']) except KeyError: self.rk = "?" try: self.rk_bnds = "%s...%s" % tuple(self.db_curves[0]['rank_bounds']) except KeyError: self.rank_bounds = [0, sage.rings.infinity.Infinity] self.rk_bnds = "not recorded" # Extract the isogeny degree matrix from the database if possible, else create it if hasattr(self, 'isogeny_matrix'): from sage.matrix.all import Matrix self.isogeny_matrix = Matrix(self.isogeny_matrix) else: self.isogeny_matrix = make_iso_matrix(self.db_curves) # Create isogeny graph: self.graph = make_graph(self.isogeny_matrix) P = self.graph.plot(edge_labels=True) self.graph_img = encode_plot(P) self.graph_link = '<img src="%s" width="200" height="150"/>' % self.graph_img self.isogeny_matrix_str = latex(matrix(self.isogeny_matrix)) self.field = field_pretty(self.field_label) self.field_knowl = nf_display_knowl(self.field_label, lmfdb.base.getDBConnection(), self.field) def curve_url(c): return url_for(".show_ecnf", nf=c['field_label'], conductor_label=c['conductor_label'], class_label=c['iso_label'], number=c['number']) self.curves = [[c['short_label'], curve_url(c), web_ainvs(self.field_label,c['ainvs'])] for c in self.db_curves] self.urls = {} self.urls['class'] = url_for(".show_ecnf_isoclass", nf=self.field_label, conductor_label=self.conductor_label, class_label=self.iso_label) self.urls['conductor'] = url_for(".show_ecnf_conductor", nf=self.field_label, conductor_label=self.conductor_label) self.urls['field'] = url_for('.show_ecnf1', nf=self.field_label) sig = self.signature real_quadratic = sig == [2,0] totally_real = sig[1] == 0 imag_quadratic = sig == [0,1] if totally_real: self.hmf_label = "-".join([self.field_label, self.conductor_label, self.iso_label]) self.urls['hmf'] = url_for('hmf.render_hmf_webpage', field_label=self.field_label, label=self.hmf_label) self.urls['Lfunction'] = url_for("l_functions.l_function_hmf_page", field=self.field_label, label=self.hmf_label, character='0', number='0') if imag_quadratic: self.bmf_label = "-".join([self.field_label, self.conductor_label, self.iso_label]) self.friends = [] if totally_real: self.friends += [('Hilbert Modular Form ' + self.hmf_label, self.urls['hmf'])] self.friends += [('L-function', self.urls['Lfunction'])] if imag_quadratic: self.friends += [('Bianchi Modular Form %s not available' % self.bmf_label, '')] self.properties = [('Base field', self.field), ('Label', self.class_label), (None, self.graph_link), ('Conductor', '%s' % self.conductor_label) ] if self.rk != '?': self.properties += [('Rank', '%s' % self.rk)] else: if self.rk_bnds == 'not recorded': self.properties += [('Rank', '%s' % self.rk_bnds)] else: self.properties += [('Rank bounds', '%s' % self.rk_bnds)] self.bread = [('Elliptic Curves ', url_for(".index")), (self.field_label, self.urls['field']), (self.conductor_label, self.urls['conductor']), ('isogeny class %s' % self.short_label, self.urls['class'])]