def nf_knowl_guts(label): out = '' wnf = WebNumberField(label) if wnf.is_null(): return 'Cannot find global number field %s' % label out += "Global number field %s" % label out += '<div>' out += 'Defining polynomial: ' out += "\(%s\)" % latex(wnf.poly()) D = wnf.disc() Dfact = wnf.disc_factored_latex() if D.abs().is_prime() or D == 1: Dfact = "\(%s\)" % str(D) else: Dfact = '%s = \(%s\)' % (str(D),Dfact) out += '<br>Discriminant: ' out += Dfact out += '<br>Signature: ' out += str(wnf.signature()) out += '<br>Galois group: '+group_pretty_and_nTj(wnf.degree(),wnf.galois_t(), True) out += '<br>Class number: %s ' % str(wnf.class_number_latex()) if wnf.can_class_number(): out += wnf.short_grh_string() out += '</div>' out += '<div align="right">' out += '<a href="%s">%s home page</a>' % (str(url_for("number_fields.number_field_render_webpage", natural=label)),label) out += '</div>' return out
def nf_knowl_guts(label): out = '' wnf = WebNumberField(label) if wnf.is_null(): return 'Cannot find global number field %s' % label out += "Global number field %s" % label out += '<div>' out += 'Defining polynomial: ' out += "\(%s\)" % latex(wnf.poly()) D = wnf.disc() Dfact = wnf.disc_factored_latex() if D.abs().is_prime() or D == 1: Dfact = "\(%s\)" % str(D) else: Dfact = '%s = \(%s\)' % (str(D), Dfact) out += '<br>Discriminant: ' out += Dfact out += '<br>Signature: ' out += str(wnf.signature()) out += '<br>Galois group: ' + group_pretty_and_nTj(wnf.degree(), wnf.galois_t(), True) out += '<br>Class number: %s ' % str(wnf.class_number_latex()) if wnf.can_class_number(): out += wnf.short_grh_string() out += '</div>' out += '<div align="right">' out += '<a href="%s">%s home page</a>' % (str( url_for("number_fields.number_field_render_webpage", natural=label)), label) out += '</div>' return out
def local_field_data(label): f = db.lf_fields.lookup(label) nicename = '' if f['n'] < 3: nicename = ' = '+ prettyname(f) ans = '$p$-adic field %s%s<br><br>'% (label, nicename) ans += r'Extension of $\Q_{%s}$ defined by %s<br>'%(str(f['p']),web_latex(coeff_to_poly(f['coeffs']))) gt = int(f['galois_label'].split('T')[1]) gn = f['n'] ans += 'Degree: %s<br>' % str(gn) ans += 'Ramification index $e$: %s<br>' % str(f['e']) ans += 'Residue field degree $f$: %s<br>' % str(f['f']) ans += 'Discriminant ideal: $(p^{%s})$ <br>' % str(f['c']) ans += 'Galois group $G$: %s<br>' % group_pretty_and_nTj(gn, gt, True) ans += '<div align="right">' ans += '<a href="%s">%s home page</a>' % (str(url_for("local_fields.by_label", label=label)),label) ans += '</div>' return ans
def local_field_data(label): f = db.lf_fields.lookup(label) nicename = '' if f['n'] < 3: nicename = ' = '+ prettyname(f) ans = 'Local number field %s%s<br><br>'% (label, nicename) ans += 'Extension of $\Q_{%s}$ defined by %s<br>'%(str(f['p']),web_latex(coeff_to_poly(f['coeffs']))) gt = f['gal'] gn = f['n'] ans += 'Degree: %s<br>' % str(gt) ans += 'Ramification index $e$: %s<br>' % str(f['e']) ans += 'Residue field degree $f$: %s<br>' % str(f['f']) ans += 'Discriminant ideal: $(p^{%s})$ <br>' % str(f['c']) ans += 'Galois group $G$: %s<br>' % group_pretty_and_nTj(gn, gt, True) ans += '<div align="right">' ans += '<a href="%s">%s home page</a>' % (str(url_for("local_fields.by_label", label=label)),label) ans += '</div>' return ans
def galois_string(self): if not self.haskey('galt'): return 'Not computed' n = self._data['degree'] t = self._data['galt'] return group_pretty_and_nTj(n, t)
def render_field_webpage(args): data = None info = {} bread = bread_prefix() # This function should not be called unless label is set. label = clean_input(args['label']) nf = WebNumberField(label) data = {} if nf.is_null(): if re.match(r'^\d+\.\d+\.\d+\.\d+$', label): flash_error("Number field %s was not found in the database.", label) else: flash_error("%s is not a valid label for a number field.", label) return redirect(url_for(".number_field_render_webpage")) info['wnf'] = nf data['degree'] = nf.degree() data['class_number'] = nf.class_number_latex() ram_primes = nf.ramified_primes() t = nf.galois_t() n = nf.degree() data['is_galois'] = nf.is_galois() data['autstring'] = r'\Gal' if data['is_galois'] else r'\Aut' data['is_abelian'] = nf.is_abelian() if nf.is_abelian(): conductor = nf.conductor() data['conductor'] = conductor dirichlet_chars = nf.dirichlet_group() if dirichlet_chars: data['dirichlet_group'] = [ r'<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for('characters.render_Dirichletwebpage', modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars ] if len(data['dirichlet_group']) == 1: data[ 'dirichlet_group'] = r'<span style="white-space:nowrap">$\lbrace$' + data[ 'dirichlet_group'][0] + r'$\rbrace$</span>' else: data['dirichlet_group'] = r'$\lbrace$' + ', '.join( data['dirichlet_group'] [:-1]) + '<span style="white-space:nowrap">' + data[ 'dirichlet_group'][-1] + r'$\rbrace$</span>' if data['conductor'].is_prime() or data['conductor'] == 1: data['conductor'] = r"\(%s\)" % str(data['conductor']) else: factored_conductor = factor_base_factor(data['conductor'], ram_primes) factored_conductor = factor_base_factorization_latex( factored_conductor) data['conductor'] = r"\(%s=%s\)" % (str( data['conductor']), factored_conductor) data['galois_group'] = group_pretty_and_nTj(n, t, True) data['auts'] = db.gps_transitive.lookup(r'{}T{}'.format(n, t))['auts'] data['cclasses'] = cclasses_display_knowl(n, t) data['character_table'] = character_table_display_knowl(n, t) data['class_group'] = nf.class_group() data['class_group_invs'] = nf.class_group_invariants() data['signature'] = nf.signature() data['coefficients'] = nf.coeffs() nf.make_code_snippets() D = nf.disc() data['disc_factor'] = nf.disc_factored_latex() if D.abs().is_prime() or D == 1: data['discriminant'] = bigint_knowl(D, cutoff=60, sides=3) else: data['discriminant'] = bigint_knowl( D, cutoff=60, sides=3) + r"\(\medspace = %s\)" % data['disc_factor'] if nf.frobs(): data['frob_data'], data['seeram'] = see_frobs(nf.frobs()) else: # fallback in case we haven't computed them in a case data['frob_data'], data['seeram'] = frobs(nf) # This could put commas in the rd, we don't want to trigger spaces data['rd'] = ('$%s$' % fixed_prec(nf.rd(), 2)).replace(',', '{,}') # Bad prime information npr = len(ram_primes) ramified_algebras_data = nf.ramified_algebras_data() if isinstance(ramified_algebras_data, str): loc_alg = '' else: # [label, latex, e, f, c, gal] loc_alg = '' for j in range(npr): if ramified_algebras_data[j] is None: loc_alg += '<tr><td>%s<td colspan="7">Data not computed' % str( ram_primes[j]).rstrip('L') else: from lmfdb.local_fields.main import show_slope_content mydat = ramified_algebras_data[j] p = ram_primes[j] loc_alg += '<tr><td rowspan="%d">$%s$</td>' % (len(mydat), str(p)) mm = mydat[0] myurl = url_for('local_fields.by_label', label=mm[0]) lab = mm[0] if mm[3] * mm[2] == 1: lab = r'$\Q_{%s}$' % str(p) loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % ( myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5], show_slope_content(mm[8], mm[6], mm[7])) for mm in mydat[1:]: lab = mm[0] myurl = url_for('local_fields.by_label', label=lab) if mm[3] * mm[2] == 1: lab = r'$\Q_{%s}$' % str(p) loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % ( myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5], show_slope_content(mm[8], mm[6], mm[7])) loc_alg += '</tbody></table>' ram_primes = str(ram_primes)[1:-1] # Get rid of python L for big numbers ram_primes = ram_primes.replace('L', '') if not ram_primes: ram_primes = r'\textrm{None}' data['phrase'] = group_phrase(n, t) zk = nf.zk() Ra = PolynomialRing(QQ, 'a') zk = [latex(Ra(x)) for x in zk] zk = ['$%s$' % x for x in zk] zk = ', '.join(zk) grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh( ) else '' # Short version for properties grh_lab = nf.short_grh_string() if 'computed' in str(data['class_number']): grh_lab = '' grh_label = '' pretty_label = field_pretty(label) if label != pretty_label: pretty_label = "%s: %s" % (label, pretty_label) info.update(data) rootofunity = '%s (order $%d$)' % (nf.root_of_1_gen(), nf.root_of_1_order()) info.update({ 'label': pretty_label, 'label_raw': label, 'polynomial': web_latex(nf.poly()), 'ram_primes': ram_primes, 'integral_basis': zk, 'regulator': web_latex(nf.regulator()), 'unit_rank': nf.unit_rank(), 'root_of_unity': rootofunity, 'fund_units': nf.units_safe(), 'cnf': nf.cnf(), 'grh_label': grh_label, 'loc_alg': loc_alg }) bread.append(('%s' % nf_label_pretty(info['label_raw']), ' ')) info['downloads_visible'] = True info['downloads'] = [('worksheet', '/')] info['friends'] = [] if nf.can_class_number(): # hide ones that take a lond time to compute on the fly # note that the first degree 4 number field missed the zero of the zeta function if abs(D**n) < 50000000: info['friends'].append(('L-function', "/L/NumberField/%s" % label)) info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t))) if 'dirichlet_group' in info: info['friends'].append(('Dirichlet character group', url_for("characters.dirichlet_group_table", modulus=int(conductor), char_number_list=','.join( str(a) for a in dirichlet_chars), poly=info['polynomial']))) resinfo = [] galois_closure = nf.galois_closure() if galois_closure[0] > 0: if galois_closure[1]: resinfo.append(('gc', galois_closure[1])) if galois_closure[2]: info['friends'].append(('Galois closure', url_for(".by_label", label=galois_closure[2][0]))) else: resinfo.append(('gc', [dnc])) sextic_twins = nf.sextic_twin() if sextic_twins[0] > 0: if sextic_twins[1]: resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1]))) else: resinfo.append(('sex', dnc)) siblings = nf.siblings() # [degsib list, label list] # first is list of [deg, num expected, list of knowls] if siblings[0]: for sibdeg in siblings[0]: if not sibdeg[2]: sibdeg[2] = dnc else: nsibs = len(sibdeg[2]) sibdeg[2] = ', '.join(sibdeg[2]) if nsibs < sibdeg[1]: sibdeg[2] += ', some ' + dnc resinfo.append(('sib', siblings[0])) for lab in siblings[1]: if lab: labparts = lab.split('.') info['friends'].append(("Degree %s sibling" % labparts[0], url_for(".by_label", label=lab))) arith_equiv = nf.arith_equiv() if arith_equiv[0] > 0: if arith_equiv[1]: resinfo.append( ('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1]))) for aelab in arith_equiv[2]: info['friends'].append(('Arithmetically equivalent sibling', url_for(".by_label", label=aelab))) else: resinfo.append(('ae', dnc, len(arith_equiv[1]))) info['resinfo'] = resinfo learnmore = learnmore_list() title = "Number field %s" % info['label'] if npr == 1: primes = 'prime' else: primes = 'primes' if len(ram_primes) > 30: ram_primes = 'see page' else: ram_primes = '$%s$' % ram_primes properties = [('Label', nf_label_pretty(label)), ('Degree', prop_int_pretty(data['degree'])), ('Signature', '$%s$' % data['signature']), ('Discriminant', prop_int_pretty(D)), ('Root discriminant', '%s' % data['rd']), ('Ramified ' + primes + '', ram_primes), ('Class number', '%s %s' % (data['class_number'], grh_lab)), ('Class group', '%s %s' % (data['class_group_invs'], grh_lab)), ('Galois group', group_pretty_and_nTj(data['degree'], t))] downloads = [('Stored data to gp', url_for('.nf_download', nf=label, download_type='data'))] for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]: downloads.append(('Download {} code'.format(lang[0]), url_for(".nf_download", nf=label, download_type=lang[1]))) from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup from lmfdb.artin_representations.math_classes import artin_label_pretty try: info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs']) arts = [ z.label() for z in info["tim_number_field"].artin_representations() ] #print arts for ar in arts: info['friends'].append(( 'Artin representation ' + artin_label_pretty(ar), url_for( "artin_representations.render_artin_representation_webpage", label=ar))) v = nf.factor_perm_repn(info["tim_number_field"]) def dopow(m): if m == 0: return '' if m == 1: return '*' return '*<sup>%d</sup>' % m info["mydecomp"] = [dopow(x) for x in v] except AttributeError: pass return render_template("nf-show-field.html", properties=properties, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), downloads=downloads, learnmore=learnmore, info=info, KNOWL_ID="nf.%s" % label)
def galois_string(self): if not self.haskey('galt'): return 'Not computed' n = self._data['degree'] t = self._data['galt'] return group_pretty_and_nTj(n, t)
def galois_string(self): if not self.haskey('galois_label'): return 'Not computed' n = self._data['degree'] t = int(self._data['galois_label'].split('T')[1]) return group_pretty_and_nTj(n, t)
def galformatter(gal): n, t = galdata(gal) return group_pretty_and_nTj(n, t, True, cache=galcache())
def render_field_webpage(args): data = None info = {} if 'label' in args: label = clean_input(args['label']) data = db.lf_fields.lookup(label) if data is None: if re.match(r'^\d+\.\d+\.\d+\.\d+$', label): flash_error("Field %s was not found in the database.", label) else: flash_error("%s is not a valid label for a $p$-adic field.", label) return redirect(url_for(".index")) title = '$p$-adic field ' + prettyname(data) titletag = 'p-adic field ' + prettyname(data) polynomial = coeff_to_poly(data['coeffs']) p = data['p'] Qp = r'\Q_{%d}' % p e = data['e'] f = data['f'] cc = data['c'] gt = int(data['galois_label'].split('T')[1]) gn = data['n'] the_gal = WebGaloisGroup.from_nt(gn, gt) isgal = ' Galois' if the_gal.order() == gn else ' not Galois' abelian = ' and abelian' if the_gal.is_abelian() else '' galphrase = 'This field is' + isgal + abelian + r' over $\Q_{%d}.$' % p autstring = r'\Gal' if the_gal.order() == gn else r'\Aut' prop2 = [ ('Label', label), ('Base', r'\(%s\)' % Qp), ('Degree', r'\(%s\)' % data['n']), ('e', r'\(%s\)' % e), ('f', r'\(%s\)' % f), ('c', r'\(%s\)' % cc), ('Galois group', group_pretty_and_nTj(gn, gt)), ] # Look up the unram poly so we can link to it unramlabel = db.lf_fields.lucky({'p': p, 'n': f, 'c': 0}, projection=0) if unramlabel is None: logger.fatal("Cannot find unramified field!") unramfriend = '' else: unramfriend = "/LocalNumberField/%s" % unramlabel unramdata = db.lf_fields.lookup(unramlabel) Px = PolynomialRing(QQ, 'x') Pt = PolynomialRing(QQ, 't') Ptx = PolynomialRing(Pt, 'x') if data['f'] == 1: unramp = r'$%s$' % Qp eisenp = Ptx(str(data['eisen']).replace('y', 'x')) eisenp = web_latex(eisenp) else: unramp = data['unram'].replace('t', 'x') unramp = web_latex(Px(str(unramp))) unramp = prettyname( unramdata ) + ' $\\cong ' + Qp + '(t)$ where $t$ is a root of ' + unramp eisenp = Ptx(str(data['eisen']).replace('y', 'x')) eisenp = '$' + web_latex(eisenp, False) + '\\in' + Qp + '(t)[x]$' rflabel = db.lf_fields.lucky( { 'p': p, 'n': { '$in': [1, 2] }, 'rf': data['rf'] }, projection=0) if rflabel is None: logger.fatal("Cannot find discriminant root field!") rffriend = '' else: rffriend = "/LocalNumberField/%s" % rflabel gsm = data['gsm'] if gsm == [0]: gsm = 'Not computed' elif gsm == [-1]: gsm = 'Does not exist' else: gsm = lf_formatfield(','.join([str(b) for b in gsm])) info.update({ 'polynomial': web_latex(polynomial), 'n': data['n'], 'p': p, 'c': data['c'], 'e': data['e'], 'f': data['f'], 't': data['t'], 'u': data['u'], 'rf': lf_display_knowl(rflabel, name=printquad(data['rf'], p)), 'base': lf_display_knowl(str(p) + '.1.0.1', name='$%s$' % Qp), 'hw': data['hw'], 'slopes': show_slopes(data['slopes']), 'gal': group_pretty_and_nTj(gn, gt, True), 'gt': gt, 'inertia': group_display_inertia(data['inertia']), 'unram': unramp, 'eisen': eisenp, 'gms': data['gms'], 'gsm': gsm, 'galphrase': galphrase, 'autstring': autstring, 'subfields': format_subfields(data['subfields'], p), 'aut': data['aut'], }) friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))] if unramfriend != '': friends.append(('Unramified subfield', unramfriend)) if rffriend != '': friends.append(('Discriminant root field', rffriend)) bread = get_bread([(label, ' ')]) return render_template( "lf-show-field.html", credit=LF_credit, title=title, titletag=titletag, bread=bread, info=info, properties=prop2, friends=friends, learnmore=learnmore_list(), KNOWL_ID="lf.%s" % label, )
def contformatter(cont): n, t = galdata(cont) if t == 0: # no T number return cont return group_pretty_and_nTj(n, t, True, cache=contcache())
def projformatter(proj): if isinstance(proj, list): proj = "%sT%s" % tuple(proj) n, t = galdata(proj) return group_pretty_and_nTj(n, t, True, cache=projcache(), skip_nTj=True)
lf_columns = SearchColumns([ LinkCol("label", "lf.field.label", "Label", url_for_label, default=True), ProcessedCol("coeffs", "lf.defining_polynomial", "Polynomial", format_coeffs, default=True), MathCol("p", "lf.qp", "$p$", default=True), MathCol("e", "lf.ramification_index", "$e$", default=True), MathCol("f", "lf.residue_field_degree", "$f$", default=True), MathCol("c", "lf.discriminant_exponent", "$c$", default=True), MultiProcessedCol( "gal", "nf.galois_group", "Galois group", ["n", "gal", "cache"], lambda n, t, cache: group_pretty_and_nTj(n, t, cache=cache), default=True), MultiProcessedCol("slopes", "lf.slope_content", "Slope content", ["slopes", "t", "u"], show_slope_content, default=True, mathmode=True) ], db_cols=[ "c", "coeffs", "e", "f", "gal", "label", "n", "p", "slopes", "t", "u" ]) def lf_postprocess(res, info, query):
def render_field_webpage(args): data = None info = {} if 'label' in args: label = clean_input(args['label']) data = db.lf_fields.lookup(label) if data is None: if re.match(r'^\d+\.\d+\.\d+\.\d+$', label): flash_error("Field <span style='color:black'>%s</span> was not found in the database.", label) else: flash_error("<span style='color:black'>%s</span> is not a valid label for a local number field.", label) return redirect(url_for(".index")) title = 'Local Number Field ' + prettyname(data) polynomial = coeff_to_poly(data['coeffs']) p = data['p'] Qp = r'\Q_{%d}' % p e = data['e'] f = data['f'] cc = data['c'] gt = data['gal'] gn = data['n'] the_gal = WebGaloisGroup.from_nt(gn,gt) isgal = ' Galois' if the_gal.order() == gn else ' not Galois' abelian = ' and abelian' if the_gal.is_abelian() else '' galphrase = 'This field is'+isgal+abelian+' over $\Q_{%d}$.'%p autstring = r'\Gal' if the_gal.order() == gn else r'\Aut' prop2 = [ ('Label', label), ('Base', '\(%s\)' % Qp), ('Degree', '\(%s\)' % data['n']), ('e', '\(%s\)' % e), ('f', '\(%s\)' % f), ('c', '\(%s\)' % cc), ('Galois group', group_pretty_and_nTj(gn, gt)), ] # Look up the unram poly so we can link to it unramlabel = db.lf_fields.lucky({'p': p, 'n': f, 'c': 0}, projection=0) if unramlabel is None: logger.fatal("Cannot find unramified field!") unramfriend = '' else: unramfriend = "/LocalNumberField/%s" % unramlabel unramdata = db.lf_fields.lookup(unramlabel) Px = PolynomialRing(QQ, 'x') Pxt=PolynomialRing(Px,'t') Pt = PolynomialRing(QQ, 't') Ptx = PolynomialRing(Pt, 'x') if data['f'] == 1: unramp = r'$%s$' % Qp # Eliminate t from the eisenstein polynomial eisenp = Pxt(str(data['eisen']).replace('y','x')) eisenp = Pt(str(data['unram'])).resultant(eisenp) eisenp = web_latex(eisenp) else: unramp = data['unram'].replace('t','x') unramp = web_latex(Px(str(unramp))) unramp = prettyname(unramdata)+' $\\cong '+Qp+'(t)$ where $t$ is a root of '+unramp eisenp = Ptx(str(data['eisen']).replace('y','x')) eisenp = '$'+web_latex(eisenp, False)+'\\in'+Qp+'(t)[x]$' rflabel = db.lf_fields.lucky({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']}, projection=0) if rflabel is None: logger.fatal("Cannot find discriminant root field!") rffriend = '' else: rffriend = "/LocalNumberField/%s" % rflabel gsm = data['gsm'] if gsm == [0]: gsm = 'Not computed' elif gsm == [-1]: gsm = 'Does not exist' else: gsm = web_latex(coeff_to_poly(gsm)) info.update({ 'polynomial': web_latex(polynomial), 'n': data['n'], 'p': p, 'c': data['c'], 'e': data['e'], 'f': data['f'], 't': data['t'], 'u': data['u'], 'rf': lf_display_knowl( rflabel, name=printquad(data['rf'], p)), 'base': lf_display_knowl(str(p)+'.1.0.1', name='$%s$'%Qp), 'hw': data['hw'], 'slopes': show_slopes(data['slopes']), 'gal': group_pretty_and_nTj(gn, gt, True), 'gt': gt, 'inertia': group_display_inertia(data['inertia']), 'unram': unramp, 'eisen': eisenp, 'gms': data['gms'], 'gsm': gsm, 'galphrase': galphrase, 'autstring': autstring, 'subfields': format_subfields(data['subfields'],p), 'aut': data['aut'], }) friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))] if unramfriend != '': friends.append(('Unramified subfield', unramfriend)) if rffriend != '': friends.append(('Discriminant root field', rffriend)) bread = get_bread([(label, ' ')]) return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore_list())
def render_field_webpage(args): data = None info = {} bread = [('Global Number Fields', url_for(".number_field_render_webpage"))] # This function should not be called unless label is set. label = clean_input(args['label']) nf = WebNumberField(label) data = {} if nf.is_null(): if re.match(r'^\d+\.\d+\.\d+\.\d+$', label): flash_error("Number field %s was not found in the database.", label) else: flash_error("%s is not a valid label for a global number field.", label) return redirect(url_for(".number_field_render_webpage")) info['wnf'] = nf data['degree'] = nf.degree() data['class_number'] = nf.class_number_latex() ram_primes = nf.ramified_primes() t = nf.galois_t() n = nf.degree() data['is_galois'] = nf.is_galois() data['is_abelian'] = nf.is_abelian() if nf.is_abelian(): conductor = nf.conductor() data['conductor'] = conductor dirichlet_chars = nf.dirichlet_group() if len(dirichlet_chars)>0: data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars] data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$' if data['conductor'].is_prime() or data['conductor'] == 1: data['conductor'] = "\(%s\)" % str(data['conductor']) else: factored_conductor = factor_base_factor(data['conductor'], ram_primes) factored_conductor = factor_base_factorization_latex(factored_conductor) data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), factored_conductor) data['galois_group'] = group_pretty_and_nTj(n,t,True) data['cclasses'] = cclasses_display_knowl(n, t) data['character_table'] = character_table_display_knowl(n, t) data['class_group'] = nf.class_group() data['class_group_invs'] = nf.class_group_invariants() data['signature'] = nf.signature() data['coefficients'] = nf.coeffs() nf.make_code_snippets() D = nf.disc() data['disc_factor'] = nf.disc_factored_latex() if D.abs().is_prime() or D == 1: data['discriminant'] = "\(%s\)" % str(D) else: data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor']) data['frob_data'], data['seeram'] = frobs(nf) # This could put commas in the rd, we don't want to trigger spaces data['rd'] = ('$%s$' % fixed_prec(nf.rd(),2)).replace(',','{,}') # Bad prime information npr = len(ram_primes) ramified_algebras_data = nf.ramified_algebras_data() if isinstance(ramified_algebras_data,str): loc_alg = '' else: # [label, latex, e, f, c, gal] loc_alg = '' for j in range(npr): if ramified_algebras_data[j] is None: loc_alg += '<tr><td>%s<td colspan="7">Data not computed'%str(ram_primes[j]).rstrip('L') else: mydat = ramified_algebras_data[j] p = ram_primes[j] loc_alg += '<tr><td rowspan="%d">$%s$</td>'%(len(mydat),str(p)) mm = mydat[0] myurl = url_for('local_fields.by_label', label=mm[0]) lab = mm[0] if mm[3]*mm[2]==1: lab = r'$\Q_{%s}$'%str(p) loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$'%(myurl,lab,mm[1],mm[2],mm[3],mm[4],mm[5],show_slope_content(mm[8],mm[6],mm[7])) for mm in mydat[1:]: lab = mm[0] if mm[3]*mm[2]==1: lab = r'$\Q_{%s}$'%str(p) loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$'%(myurl,lab,mm[1],mm[2],mm[3],mm[4],mm[5],show_slope_content(mm[8],mm[6],mm[7])) loc_alg += '</tbody></table>' ram_primes = str(ram_primes)[1:-1] # Get rid of python L for big numbers ram_primes = ram_primes.replace('L', '') if ram_primes == '': ram_primes = r'\textrm{None}' data['phrase'] = group_phrase(n, t) zk = nf.zk() Ra = PolynomialRing(QQ, 'a') zk = [latex(Ra(x)) for x in zk] zk = ['$%s$' % x for x in zk] zk = ', '.join(zk) grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else '' # Short version for properties grh_lab = nf.short_grh_string() if 'Not' in str(data['class_number']): grh_lab='' grh_label='' pretty_label = field_pretty(label) if label != pretty_label: pretty_label = "%s: %s" % (label, pretty_label) info.update(data) if nf.degree() > 1: gpK = nf.gpK() rootof1coeff = gpK.nfrootsof1() rootofunityorder = int(rootof1coeff[1]) rootof1coeff = rootof1coeff[2] rootofunity = web_latex(Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a'))) rootofunity += ' (order $%d$)' % rootofunityorder else: rootofunity = web_latex(Ra('-1'))+ ' (order $2$)' info.update({ 'label': pretty_label, 'label_raw': label, 'polynomial': web_latex_split_on_pm(nf.poly()), 'ram_primes': ram_primes, 'integral_basis': zk, 'regulator': web_latex(nf.regulator()), 'unit_rank': nf.unit_rank(), 'root_of_unity': rootofunity, 'fund_units': nf.units(), 'grh_label': grh_label, 'loc_alg': loc_alg }) bread.append(('%s' % info['label_raw'], ' ')) info['downloads_visible'] = True info['downloads'] = [('worksheet', '/')] info['friends'] = [] if nf.can_class_number(): # hide ones that take a lond time to compute on the fly # note that the first degree 4 number field missed the zero of the zeta function if abs(D**n) < 50000000: info['friends'].append(('L-function', "/L/NumberField/%s" % label)) info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t))) if 'dirichlet_group' in info: info['friends'].append(('Dirichlet character group', url_for("characters.dirichlet_group_table", modulus=int(conductor), char_number_list=','.join( [str(a) for a in dirichlet_chars]), poly=info['polynomial']))) resinfo=[] galois_closure = nf.galois_closure() if galois_closure[0]>0: if len(galois_closure[1])>0: resinfo.append(('gc', galois_closure[1])) if len(galois_closure[2]) > 0: info['friends'].append(('Galois closure',url_for(".by_label", label=galois_closure[2][0]))) else: resinfo.append(('gc', [dnc])) sextic_twins = nf.sextic_twin() if sextic_twins[0]>0: if len(sextic_twins[1])>0: resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1]))) else: resinfo.append(('sex', dnc)) siblings = nf.siblings() # [degsib list, label list] # first is list of [deg, num expected, list of knowls] if len(siblings[0])>0: for sibdeg in siblings[0]: if len(sibdeg[2]) ==0: sibdeg[2] = dnc else: sibdeg[2] = ', '.join(sibdeg[2]) if len(sibdeg[2])<sibdeg[1]: sibdeg[2] += ', some '+dnc resinfo.append(('sib', siblings[0])) for lab in siblings[1]: if lab != '': labparts = lab.split('.') info['friends'].append(("Degree %s sibling"%labparts[0] ,url_for(".by_label", label=lab))) arith_equiv = nf.arith_equiv() if arith_equiv[0]>0: if len(arith_equiv[1])>0: resinfo.append(('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1]))) for aelab in arith_equiv[2]: info['friends'].append(('Arithmetically equivalent sibling',url_for(".by_label", label=aelab))) else: resinfo.append(('ae', dnc, len(arith_equiv[1]))) info['resinfo'] = resinfo learnmore = learnmore_list() title = "Global Number Field %s" % info['label'] if npr == 1: primes = 'prime' else: primes = 'primes' if len(label)>25: label = label[:16]+'...'+label[-6:] properties2 = [('Label', label), ('Degree', '$%s$' % data['degree']), ('Signature', '$%s$' % data['signature']), ('Discriminant', '$%s$' % data['disc_factor']), ('Root discriminant', '%s' % data['rd']), ('Ramified ' + primes + '', '$%s$' % ram_primes), ('Class number', '%s %s' % (data['class_number'], grh_lab)), ('Class group', '%s %s' % (data['class_group_invs'], grh_lab)), ('Galois Group', group_pretty_and_nTj(data['degree'], t)) ] downloads = [] for lang in [["Magma","magma"], ["SageMath","sage"], ["Pari/GP", "gp"]]: downloads.append(('Download {} code'.format(lang[0]), url_for(".nf_code_download", nf=label, download_type=lang[1]))) from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup try: info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs']) v = nf.factor_perm_repn(info["tim_number_field"]) def dopow(m): if m==0: return '' if m==1: return '*' return '*<sup>%d</sup>'% m info["mydecomp"] = [dopow(x) for x in v] except AttributeError: pass return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), downloads=downloads, learnmore=learnmore, info=info, KNOWL_ID="nf.%s"%label)