def row(trclass, goodorbad, p, poly):
out = ""
try:
if L.coefficient_field == "CDF" or None in poly:
factors = str(pretty_poly(poly, prec = prec))
elif not display_galois:
factors = list_to_factored_poly_otherorder(poly, galois=display_galois, prec = prec, p = p)
else:
factors, gal_groups = list_to_factored_poly_otherorder(poly, galois=display_galois, p = p)
out += "<tr" + trclass + "><td>" + goodorbad + "</td><td>" + str(p) + "</td>";
if display_galois:
out += "<td class='galois'>"
if gal_groups[0]==[0,0]:
pass # do nothing, because the local faco is 1
elif gal_groups[0]==[1,1]:
out += group_display_knowl(gal_groups[0][0], gal_groups[0][1],'$C_1$')
else:
out += group_display_knowl(gal_groups[0][0], gal_groups[0][1])
for n, k in gal_groups[1:]:
out += "$\\times$"
out += group_display_knowl(n, k)
out += "</td>"
out += "<td>" +"$" + factors + "$" + "</td>"
out += "</tr>\n"
except IndexError:
out += "<tr><td></td><td>" + str(j) + "</td><td>" + "not available" + "</td></tr>\n"
return out
def row(trclass, goodorbad, p, poly):
out = ""
try:
if L.coefficient_field == "CDF" or None in poly:
factors = str(pretty_poly(poly, prec=prec))
elif not display_galois:
factors = list_to_factored_poly_otherorder(
poly, galois=display_galois, prec=prec, p=p)
else:
factors, gal_groups = list_to_factored_poly_otherorder(
poly, galois=display_galois, p=p)
out += "<tr" + trclass + "><td>" + goodorbad + "</td><td>" + str(
p) + "</td>"
if display_galois:
out += "<td class='galois'>"
if gal_groups[0] == [0, 0]:
pass # do nothing, because the local faco is 1
elif gal_groups[0] == [1, 1]:
out += group_display_knowl(gal_groups[0][0],
gal_groups[0][1], '$C_1$')
else:
out += group_display_knowl(gal_groups[0][0],
gal_groups[0][1])
for n, k in gal_groups[1:]:
out += "$\\times$"
out += group_display_knowl(n, k)
out += "</td>"
out += "<td>" + "$" + factors + "$" + "</td>"
out += "</tr>\n"
except IndexError:
out += "<tr><td></td><td>" + str(
j) + "</td><td>" + "not available" + "</td></tr>\n"
return out
def display_galois_group(self):
if not self.gal or not self.gal[
't']: #the number field was not found in the database
return "The Galois group of this isogeny class is not in the database."
else:
C = getDBConnection()
return group_display_knowl(self.gal['n'], self.gal['t'], C)
def get_local_algebra(self, p):
local_algebra_dict = self._data.get('loc_algebras', None)
if local_algebra_dict is None:
return None
if str(p) in local_algebra_dict:
R = PolynomialRing(QQ, 'x')
palg = local_algebra_dict[str(p)]
palgs = [R(str(s)) for s in palg.split(',')]
palgstr = [
list2string([int(c) for c in pol.coefficients(sparse=False)])
for pol in palgs]
palgrec = [db.lf_fields.lucky({'p': p, 'coeffs': map(int, c.split(','))}) for c in palgstr]
return [
[
LF['label'],
latex(f),
int(LF['e']),
int(LF['f']),
int(LF['c']),
group_display_knowl(LF['n'], LF['galT']),
LF['t'],
LF['u'],
LF['slopes']
]
for LF, f in zip(palgrec, palgs) ]
return None
def display_galois_group(self):
if not hasattr(
self, 'galois_t'
) or not self.galois_t: #the number field was not found in the database
return "The Galois group of this isogeny class is not in the database."
else:
return group_display_knowl(self.galois_n, self.galois_t)
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_display_knowl(wnf.degree(),
wnf.galois_t())
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 get_local_algebra(self, p):
local_algebra_dict = self._data.get('loc_algebras', None)
if local_algebra_dict is None:
return None
if str(p) in local_algebra_dict:
R = PolynomialRing(QQ, 'x')
palg = local_algebra_dict[str(p)]
palgs = [R(str(s)) for s in palg.split(',')]
palgstr = [
list2string([int(c) for c in pol.coefficients(sparse=False)])
for pol in palgs
]
palgrec = [
db.lf_fields.lucky({
'p': p,
'coeffs': map(int, c.split(','))
}) for c in palgstr
]
return [[
LF['label'],
latex(f),
int(LF['e']),
int(LF['f']),
int(LF['c']),
group_display_knowl(LF['n'], LF['galT']), LF['t'], LF['u'],
LF['slopes']
] for LF, f in zip(palgrec, palgs)]
return None
def nf_knowl_guts(label, C):
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_display_knowl(wnf.degree(),wnf.galois_t(),C)
out += '</div>'
out += '<div align="right">'
out += '<a href="%s">%s home page</a>' % (url_for("number_fields.number_field_render_webpage", natural=label),label)
out += '</div>'
return out
def av_data(label):
abvar = db.av_fqisog.lookup(label)
wnf = WebNumberField(abvar['nf'])
inf = '<div>Dimension: ' + str(abvar['g']) + '<br />'
if not wnf.is_null():
inf += 'Number field: ' + nf_display_knowl(abvar['nf'], name = abvar['nf']) + '<br />'
inf += 'Galois group: ' + group_display_knowl(abvar['gal']['n'],abvar['gal']['t']) + '<br />'
inf += '$p$-rank: ' + str(abvar['p_rank']) + '</div>'
inf += '<div align="right">'
g, q, iso = split_label(label)
url = url_for("abvarfq.abelian_varieties_by_gqi", g = g, q = q, iso = iso)
inf += '<a href="%s">%s home page</a>' % (url, label)
inf += '</div>'
return inf
def lf_knowl_guts(label, C):
f = C.localfields.fields.find_one({'label':label})
ans = 'Local number field %s<br><br>'% label
ans += 'Extension of $\Q_{%s}$ defined by %s<br>'%(str(f['p']),web_latex(coeff_to_poly(string2list(f['coeffs']))))
GG = f['gal']
ans += 'Degree: %s<br>' % str(GG[0])
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_display_knowl(GG[0], GG[1], C)
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 lf_knowl_guts(label, C):
f = C.localfields.fields.find_one({'label':label})
ans = 'Local number field %s<br><br>'% label
ans += 'Extension of $\Q_{%s}$ defined by %s<br>'%(str(f['p']),web_latex(coeff_to_poly(string2list(f['coeffs']))))
GG = f['gal']
ans += 'Degree: %s<br>' % str(GG[0])
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_display_knowl(GG[0], GG[1], C)
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 lf_knowl_guts(label, C):
f = C.localfields.fields.find_one({"label": label})
ans = "Local number field %s<br><br>" % label
ans += "Extension of $\Q_{%s}$ defined by %s<br>" % (str(f["p"]), web_latex(coeff_to_poly(f["coeffs"])))
GG = f["gal"]
ans += "Degree: %s<br>" % str(GG[0])
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_display_knowl(GG[0], GG[1], C)
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 av_data(label):
C = getDBConnection()
abvar = C.abvar.fq_isog.find_one({ 'label' : label })
wnf = WebNumberField(abvar['nf'])
inf = '<div>Dimension: ' + str(abvar['g']) + '<br />'
if not wnf.is_null():
inf += 'Number field: ' + nf_display_knowl(abvar['nf'], C, name = abvar['nf']) + '<br />'
inf += 'Galois group: ' + group_display_knowl(abvar['gal']['n'],abvar['gal']['t'],C) + '<br />'
inf += '$p$-rank: ' + str(abvar['p_rank']) + '</div>'
inf += '<div align="right">'
g, q, iso = split_label(label)
url = url_for("abvarfq.abelian_varieties_by_gqi", g = g, q = q, iso = iso)
inf += '<a href="%s">%s home page</a>' % (url, label)
inf += '</div>'
return inf
def get_local_algebra(self, p):
local_algebra_dict = self._data.get('loc_algebras', None)
if local_algebra_dict is None:
return None
if str(p) in local_algebra_dict:
R = PolynomialRing(QQ, 'x')
palg = local_algebra_dict[str(p)]
palgs = [R(str(s)) for s in palg.split(',')]
palgs = [list2string([int(c) for c in
pol.coefficients(sparse=False)]) for pol in palgs]
palgs = [lfdb().find_one({'p': p, 'coeffs': c}) for c in palgs]
return [[f['label'], latex(R(string2list(f['coeffs']))),
int(f['e']), int(f['f']),int(f['c']),
group_display_knowl(f['gal'][0], f['gal'][1], db()),
f['t'],f['u'],f['slopes']]
for f in palgs]
return None
def get_local_algebra(self, p):
local_algebra_dict = self._data.get('loc_algebras', None)
if local_algebra_dict is None:
return None
if str(p) in local_algebra_dict:
R = PolynomialRing(QQ, 'x')
palg = local_algebra_dict[str(p)]
palgs = [R(str(s)) for s in palg.split(',')]
palgs = [list2string([int(c) for c in
pol.coefficients(sparse=False)]) for pol in palgs]
palgs = [lfdb().find_one({'p': p, 'coeffs': c}) for c in palgs]
return [[f['label'], latex(R(string2list(f['coeffs']))),
int(f['e']), int(f['f']),int(f['c']),
group_display_knowl(f['gal'][0], f['gal'][1], db()),
f['t'],f['u'],f['slopes']]
for f in palgs]
return None
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_display_knowl(gn, gt)
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 lf_algebra_knowl_guts(labels, C):
labs = labels.split(',')
f1 = labs[0].split('.')
labs = sorted(labs, key=lambda u: (int(j) for j in u.split('.')), reverse=True)
ans = '<div align="center">'
ans += '$%s$-adic algebra'%str(f1[0])
ans += '</div>'
ans += '<p>'
ans += "<table class='ntdata'><th>Label<th>Polynomial<th>$e$<th>$f$<th>$c$<th>$G$<th>Slopes"
fall = [C.localfields.fields.find_one({'label':label}) for label in labs]
for f in fall:
l = str(f['label'])
ans += '<tr><td><a href="/LocalNumberField/%s">%s</a><td>'%(l,l)
ans += format_coeffs(f['coeffs'])
ans += '<td>%d<td>%d<td>%d<td>'%(f['e'],f['f'],f['c'])
ans += group_display_knowl(f['gal'][0],f['gal'][1],db())
ans += '<td>$'+ show_slope_content(f['slopes'],f['t'],f['u'])+'$'
ans += '</table>'
if len(labs) != len(set(labs)):
ans +='<p>Fields which appear more than once occur according to their given multiplicities in the algebra'
return ans
def lf_algebra_knowl_guts(labels, C):
labs = labels.split(',')
f1 = labs[0].split('.')
labs = sorted(labs, key=lambda u: (int(j) for j in u.split('.')), reverse=True)
ans = '<div align="center">'
ans += '$%s$-adic algebra'%str(f1[0])
ans += '</div>'
ans += '<p>'
ans += "<table class='ntdata'><th>Label<th>Polynomial<th>$e$<th>$f$<th>$c$<th>$G$<th>Slopes"
fall = [C.localfields.fields.find_one({'label':label}) for label in labs]
for f in fall:
l = str(f['label'])
ans += '<tr><td><a href="/LocalNumberField/%s">%s</a><td>'%(l,l)
ans += format_coeffs(f['coeffs'])
ans += '<td>%d<td>%d<td>%d<td>'%(f['e'],f['f'],f['c'])
ans += group_display_knowl(f['gal'][0],f['gal'][1],db())
ans += '<td>$'+ show_slope_content(f['slopes'],f['t'],f['u'])+'$'
ans += '</table>'
if len(labs) != len(set(labs)):
ans +='<p>Fields which appear more than once occur according to their given multiplicities in the algebra'
return ans
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label})
bread = get_bread([(label, ' ')])
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(
url_for('.render_artin_representation_webpage', label=clean_label),
301)
try:
the_rep = ArtinRepresentation(label)
except:
flash(
Markup(
"Error: <span style='color:black'>%s</span> is not the label of an Artin representation in the database."
% (label)), "error")
return search_input_error({'err': ''}, bread)
extra_data = {} # for testing?
extra_data['galois_knowl'] = group_display_knowl(5, 3) # for testing?
#artin_logger.info("Found %s" % (the_rep._data))
title = "Artin Representation %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [
("Label", label),
("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
#("Conductor", str(the_rep.conductor())),
("Conductor", "$" + the_rep.factored_conductor_latex() + "$"),
#("Bad primes", str(the_rep.bad_primes())),
("Root number", processed_root_number),
("Frobenius-Schur indicator", str(the_rep.indicator()))
]
friends = []
nf_url = the_nf.url_for()
if nf_url:
friends.append(("Artin Field", nf_url))
cc = the_rep.central_character()
if cc is not None:
if the_rep.dimension() == 1:
if cc.order == 2:
cc_name = cc.symbol
else:
cc_name = cc.texname
friends.append(("Dirichlet character " + cc_name,
url_for("characters.render_Dirichletwebpage",
modulus=cc.modulus,
number=cc.number)))
else:
detrep = the_rep.central_character_as_artin_rep()
friends.append(("Determinant representation " + detrep.label(),
detrep.url_for()))
# once the L-functions are in the database, the link can always be shown
#if the_rep.dimension() <= 6:
if the_rep.dimension() == 1:
# Zeta is loaded differently
if cc.modulus == 1 and cc.number == 1:
friends.append(("L-function",
url_for("l_functions.l_function_dirichlet_page",
modulus=cc.modulus,
number=cc.number)))
else:
# looking for Lhash dirichlet_L_modulus.number
mylhash = 'dirichlet_L_%d.%d' % (cc.modulus, cc.number)
lres = db.lfunc_instances.lucky({'Lhash': mylhash})
if lres is not None:
friends.append(
("L-function",
url_for("l_functions.l_function_dirichlet_page",
modulus=cc.modulus,
number=cc.number)))
# Dimension > 1
elif int(the_rep.conductor())**the_rep.dimension() <= 729000000000000:
friends.append(("L-function",
url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
info = {}
#mychar = the_rep.central_char()
#info['pol2']= str([((j+1),mychar(j+1, 2*the_rep.character_field())) for j in range(50)])
#info['pol3']=str(the_rep.central_character())
#info['pol3']=str(the_rep.central_char(3))
#info['pol5']=str(the_rep.central_char(5))
#info['pol7']=str(the_rep.central_char(7))
#info['pol11']=str(the_rep.central_char(11))
learnmore = [('Artin representations labels', url_for(".labels_page"))]
return render_template("artin-representation-show.html",
credit=tim_credit,
support=support_credit,
title=title,
bread=bread,
friends=friends,
object=the_rep,
properties2=properties,
extra_data=extra_data,
info=info,
learnmore=learnmore)
def make_object(self, curve, endo, tama, ratpts, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = curve['Lhash']
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex(factor(int(data['cond'])))
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1,4,data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page", cond=data['slabel'][0], x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['disc_key'][3:]) # use disc_key rather than abs_disc (will work when abs_disc > 2^63)
data['disc'] = curve['disc_sign'] * curve['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = list_to_min_eqn(data['min_eqn'])
data['disc_factor_latex'] = web_latex(factor(data['disc']))
data['igusa_clebsch'] = [ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [web_latex(zfactor(i)) for i in data['igusa_clebsch']]
data['igusa_factor_latex'] = [ web_latex(zfactor(j)) for j in data['igusa'] ]
data['aut_grp_id'] = curve['aut_grp_id']
data['geom_aut_grp_id'] = curve['geom_aut_grp_id']
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['has_square_sha'] = "square" if curve['has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",",1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['torsion_order'] = curve['torsion_order']
data['torsion_factors'] = [ ZZ(a) for a in literal_eval(curve['torsion_subgroup']) ]
if len(data['torsion_factors']) == 0:
data['torsion_subgroup'] = '\mathrm{trivial}'
else:
data['torsion_subgroup'] = ' \\times '.join([ '\Z/{%s}\Z' % n for n in data['torsion_factors'] ])
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['tama'] = ''
for i in range(tama.count()):
item = tama.next()
if item['tamagawa_number'] > 0:
tamgwnr = str(item['tamagawa_number'])
else:
tamgwnr = 'N/A'
data['tama'] += tamgwnr + ' (p = ' + str(item['p']) + ')'
if (i+1 < tama.count()):
data['tama'] += ', '
if ratpts:
if len(ratpts['rat_pts']):
data['rat_pts'] = ', '.join(web_latex('(' +' : '.join(P) + ')') for P in ratpts['rat_pts'])
data['rat_pts_v'] = 2 if ratpts['rat_pts_v'] else 1
# data['mw_rank'] = ratpts['mw_rank']
# data['mw_rank_v'] = ratpts['mw_rank_v']
else:
data['rat_pts_v'] = 0
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl (curve['two_torsion_field'][0], getDBConnection(), field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data['two_torsion_field_knowl'] = """splitting field of \(%s\) with Galois group %s"""%(intlist_to_poly(t[1]),group_display_knowl(t[2][0],t[2][1],getDBConnection()))
else:
# invariants specific to isogeny class
curves_data = g2c_db_curves().find({"class" : curve['class']},{'_id':int(0),'label':int(1),'eqn':int(1),'disc_key':int(1)}).sort([("disc_key", ASCENDING), ("label", ASCENDING)])
if not curves_data:
raise KeyError("No curves found in database for isogeny class %s of genus 2 curve %s." %(curve['class'],curve['label']))
data['curves'] = [ {"label" : c['label'], "equation_formatted" : list_to_min_eqn(literal_eval(c['eqn'])), "url": url_for_curve_label(c['label'])} for c in curves_data ]
lfunc_data = g2c_db_lfunction_by_hash(curve['Lhash'])
if not lfunc_data:
raise KeyError("No Lfunction found in database for isogeny class of genus 2 curve %s." %curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [[nth_prime(n+1),lfunc_data['euler_factors'][n]] for n in range(len(lfunc_data['euler_factors'])) if nth_prime(n+1) < 30 and (data['cond'] % nth_prime(n+1))]
data['good_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['good_lfactors']]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = """Endomorphism %s over \(\Q\):<br>""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_base'], endo['factorsRR_base'], ring=data['end_ring_base'] if is_curve else None)
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = """Endomorphism %s over \(\overline{\Q}\):""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_geom'], data['factorsRR_geom'], field=r'\overline{\Q}', ring=data['end_ring_geom'] if is_curve else None)
data['real_geom_end_alg_name'] = end_alg_name(curve['real_geom_end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(data['is_simple_geom'], data['split_field_label'], data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'], data.get('split_labels'), data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
self.plot = encode_plot(eqn_list_to_curve_plot(data['min_eqn'], data['rat_pts'].split(',') if 'rat_pts' in data else []))
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(self.plot)
properties += [
(None, plot_link),
('Conductor',str(data['cond'])),
('Discriminant', str(data['disc'])),
]
properties += [
('Sato-Tate group', data['st_group_link']),
('\(\\End(J_{\\overline{\\Q}}) \\otimes \\R\)', '\(%s\)' % data['real_geom_end_alg_name']),
('\(\\overline{\\Q}\)-simple', bool_pretty(data['is_simple_geom'])),
('\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = [('L-function', data['lfunc_url'])]
if is_curve:
friends.append(('Isogeny class %s.%s' % (data['slabel'][0], data['slabel'][1]), url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1])))
for friend in g2c_db_lfunction_instances().find({'Lhash':data['Lhash']},{'_id':False,'url':True}):
if 'url' in friend:
add_friend (friends, lfunction_friend_from_url(friend['url']))
if 'urls' in friend:
for url in friends['urls']:
add_friend (friends, lfunction_friend_from_url(friend['url']))
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
add_friend (friends, ("Elliptic curve " + friend_label, url_for_ec(friend_label)))
else:
add_friend (friends, ("EC isogeny class " + ec_label_class(friend_label), url_for_ec_class(friend_label)))
if is_curve:
friends.append(('Twists', url_for(".index_Q", g20 = str(data['g2'][0]), g21 = str(data['g2'][1]), g22 = str(data['g2'][2]))))
# Breadcrumbs
self.bread = bread = [
('Genus 2 Curves', url_for(".index")),
('$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0], url_for(".by_conductor", cond=data['slabel'][0])),
('%s' % data['slabel'][1], url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1]))
]
if is_curve:
bread += [
('%s' % data['slabel'][2], url_for(".by_url_isogeny_class_discriminant", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2])),
('%s' % data['slabel'][3], url_for(".by_url_curve_label", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2], num=data['slabel'][3]))
]
# Title
self.title = "Genus 2 " + ("Curve " if is_curve else "Isogeny Class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage':'','magma':''} # use default show names
code['curve'] = {'sage':'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s))'%(data['min_eqn'][0],data['min_eqn'][1]),
'magma':'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'%(data['min_eqn'][0],data['min_eqn'][1])}
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation('+str(data['cond'])+',2),R!'+str(bad2)+'>*]'
else:
magma_cond_option = ''
code['cond'] = {'magma': 'Conductor(LSeries(C%s)); Factorization($1);'% magma_cond_option}
code['disc'] = {'magma':'Discriminant(C); Factorization(Integers()!$1);'}
code['igusa_clebsch'] = {'sage':'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':'IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];'}
code['igusa'] = {'magma':'IgusaInvariants(C); [Factorization(Integers()!a): a in $1];'}
code['g2'] = {'magma':'G2Invariants(C);'}
code['aut'] = {'magma':'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {'magma':'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'}
code['num_rat_wpts'] = {'magma':'#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'}
if ratpts:
code['rat_pts'] = {'magma': '[' + ','.join(["C![%s,%s,%s]"%(p[0],p[1],p[2]) for p in ratpts['rat_pts']]) + '];' }
code['two_selmer'] = {'magma':'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'}
code['has_square_sha'] = {'magma':'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {'magma':'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'}
code['torsion_subgroup'] = {'magma':'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'}
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = lfdb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field ' + label
polynomial = coeff_to_poly(string2list(data['coeffs']))
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
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', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, db())),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = lfdb().find_one({'p': p, 'n': f, 'c': 0})
if unramdata is not None:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = lfdb().find_one({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
gsm = data['gsm']
if gsm == '0':
gsm = 'Not computed'
elif gsm == '-1':
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(string2list(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': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, db()),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], db()),
'unram': web_latex(unramp),
'eisen': web_latex(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, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
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():
bread.append(('Search Results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
info['label'] = args['label_orig'] if 'label_orig' in args else args[
'label']
return search_input_error(info, bread)
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_display_knowl(n, t)
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)
# 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])
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]
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()
#if info['signature'] == [0,1]:
# info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Label', label), ('Degree', '$%s$' % data['degree']),
('Signature', '$%s$' % data['signature']),
('Discriminant', '$%s$' % data['disc_factor']),
('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_display_short(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)
def pretty_galois_knowl(self):
C = getDBConnection()
return group_display_knowl(self._data["Galois_nt"][0], self._data["Galois_nt"][1], C)
def smallest_gal_t_format(self):
galnt = self.smallest_gal_t()
if len(galnt)==1:
return galnt[0]
C = getDBConnection()
return group_display_knowl(galnt[0],galnt[1],C)
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.localfields.fields.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field ' + label
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
prop2 = [
('Label', label),
('Base', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, C)),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = C.localfields.fields.find_one({'p': p, 'n': f, 'c': 0})
if len(unramdata) > 0:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = C.localfields.fields.find_one({
'p': p,
'n': {
'$in': [1, 2]
},
'rf': data['rf']
})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': data['p'],
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, C),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], C),
'unram': web_latex(unramp),
'eisen': web_latex(eisenp),
'gms': data['gms'],
'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, ' ')])
learnmore = [('Completeness of the data',
url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html",
credit=LF_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore)
def make_passport_object(self, passport):
from lmfdb.belyi.main import url_for_belyi_galmap_label
# all information about the map goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
for elt in ('plabel', 'abc', 'num_orbits', 'g', 'abc', 'deg', 'maxdegbf'):
data[elt] = passport[elt]
nt = passport['group'].split('T')
data['group'] = group_display_knowl(int(nt[0]),int(nt[1]))
data['geomtype'] = geomtypelet_to_geomtypename_dict[passport['geomtype']]
data['lambdas'] = [str(c)[1:-1] for c in passport['lambdas']]
data['pass_size'] = passport['pass_size']
# Permutation triples
galmaps_for_plabel = db.belyi_galmaps.search( {"plabel" : passport['plabel']})#, sort = ['label_index'])
galmapdata = []
for galmap in galmaps_for_plabel:
# wrap number field nonsense
F = belyi_base_field(galmap)
# inLMFDB = False;
field = {};
if F._data == None:
field['in_LMFDB'] = False;
fld_coeffs = galmap['base_field']
pol = PolynomialRing(QQ, 'x')(fld_coeffs)
field['base_field'] = latex(pol)
field['isQQ'] = False;
else:
field['in_LMFDB'] = True;
if F.poly().degree()==1:
field['isQQ'] = True
F.latex_poly = web_latex(F.poly())
field['base_field'] = F
galmapdatum = [galmap['label'].split('-')[-1],
url_for_belyi_galmap_label(galmap['label']),
galmap['orbit_size'],
field,
galmap['triples_cyc'][0][0],
galmap['triples_cyc'][0][1],
galmap['triples_cyc'][0][2],
]
galmapdata.append(galmapdatum)
data['galmapdata'] = galmapdata
# Properties
properties = [('Label', passport['plabel']),
('Group', str(passport['group'])),
('Orders', str(passport['abc'])),
('Genus', str(passport['g'])),
('Size', str(passport['pass_size'])),
('Galois orbits', str(passport['num_orbits']))
]
self.properties = properties
# Friends
self.friends = []
# Breadcrumbs
groupstr, abcstr, sigma0, sigma1, sigmaoo, gstr = data['plabel'].split("-");
lambdasstr = '%s-%s-%s' % (sigma0, sigma1, sigmaoo);
lambdasgstr = lambdasstr + "-" + gstr;
self.bread = [
('Belyi Maps', url_for(".index")),
(groupstr,
url_for(".by_url_belyi_search_group",
group=groupstr
)
),
(abcstr,
url_for(".by_url_belyi_search_group_triple",
group=groupstr,
abc=abcstr
)
),
(lambdasgstr,
url_for(".by_url_belyi_passport_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g = gstr )
)
];
# Title
self.title = "Passport " + data['plabel']
# Code snippets (only for curves)
self.code = {}
return
def make_galmap_object(self, galmap):
from lmfdb.belyi.main import url_for_belyi_passport_label
# all information about the map goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
# the stuff that does not need to be polished
for elt in ('label', 'plabel', 'triples_cyc', 'orbit_size', 'g', 'abc', 'deg'):
data[elt] = galmap[elt]
nt = galmap['group'].split('T')
data['group'] = group_display_knowl(int(nt[0]),int(nt[1]))
data['geomtype'] = geomtypelet_to_geomtypename_dict[galmap['geomtype']]
data['lambdas'] = [str(c)[1:-1] for c in galmap['lambdas']]
data['isQQ'] = False
data['in_LMFDB'] = False
F = belyi_base_field(galmap)
if F._data == None:
fld_coeffs = galmap['base_field']
pol = PolynomialRing(QQ, 'x')(fld_coeffs)
data['base_field'] = latex(pol)
else:
data['in_LMFDB'] = True
if F.poly().degree()==1:
data['isQQ'] = True
F.latex_poly = web_latex(F.poly())
data['base_field'] = F
crv_str = galmap['curve']
if crv_str=='PP1':
data['curve'] = '\mathbb{P}^1'
else:
data['curve'] = make_curve_latex(crv_str)
# change pairs of floats to complex numbers
embeds = galmap['embeddings']
embed_strs = []
for el in embeds:
if el[1] < 0:
el_str = str(el[0]) + str(el[1]) + "\sqrt{-1}"
else:
el_str = str(el[0]) + "+" + str(el[1]) + "\sqrt{-1}"
embed_strs.append(el_str)
data['map'] = make_map_latex(galmap['map'])
data['embeddings_and_triples'] = []
if data['isQQ']:
for i in range(0,len(data['triples_cyc'])):
triple_cyc = data['triples_cyc'][i]
data['embeddings_and_triples'].append(["\\text{not applicable (over $\mathbb{Q}$)}", triple_cyc[0], triple_cyc[1], triple_cyc[2]])
else:
for i in range(0,len(data['triples_cyc'])):
triple_cyc = data['triples_cyc'][i]
data['embeddings_and_triples'].append([embed_strs[i], triple_cyc[0], triple_cyc[1], triple_cyc[2]])
data['lambdas'] = [str(c)[1:-1] for c in galmap['lambdas']]
# Properties
properties = [('Label', galmap['label']),
('Group', str(galmap['group'])),
('Orders', str(galmap['abc'])),
('Genus', str(galmap['g'])),
('Size', str(galmap['orbit_size'])),
]
self.properties = properties
# Friends
self.friends = [('Passport', url_for_belyi_passport_label(galmap['plabel']))]
# Breadcrumbs
groupstr, abcstr, sigma0, sigma1, sigmaoo, gstr, letnum = data['label'].split("-");
lambdasstr = '%s-%s-%s' % (sigma0, sigma1, sigmaoo);
lambdasgstr = lambdasstr + "-" + gstr;
self.bread = [
('Belyi Maps', url_for(".index")),
(groupstr,
url_for(".by_url_belyi_search_group",
group=groupstr
)
),
(abcstr,
url_for(".by_url_belyi_search_group_triple",
group=groupstr,
abc=abcstr
)
),
(lambdasgstr,
url_for(".by_url_belyi_passport_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g = gstr )
),
(letnum,
url_for(".by_url_belyi_galmap_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g = gstr,
letnum = letnum)
),
];
# Title
self.title = "Belyi map " + data['label']
# Code snippets (only for curves)
self.code = {}
return
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():
bread.append(('Search Results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
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_display_knowl(n, t)
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)
# 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])
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]
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()
#if info['signature'] == [0,1]:
# info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Label', label),
('Degree', '$%s$' % data['degree']),
('Signature', '$%s$' % data['signature']),
('Discriminant', '$%s$' % data['disc_factor']),
('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_display_short(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)
def make_passport_object(self, passport):
from lmfdb.belyi.main import url_for_belyi_galmap_label
# all information about the map goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
for elt in ('plabel', 'abc', 'num_orbits', 'g', 'abc', 'deg',
'maxdegbf'):
data[elt] = passport[elt]
nt = passport['group'].split('T')
data['group'] = group_display_knowl(int(nt[0]), int(nt[1]),
getDBConnection())
data['geomtype'] = geomtypelet_to_geomtypename_dict[
passport['geomtype']]
data['lambdas'] = [str(c)[1:-1] for c in passport['lambdas']]
data['pass_size'] = passport['pass_size']
# Permutation triples
galmaps_for_plabel = belyi_db_galmaps().find({
"plabel":
passport['plabel']
}).sort([('label_index', ASCENDING)])
galmapdata = []
for galmap in galmaps_for_plabel:
# wrap number field nonsense
F = belyi_base_field(galmap)
# inLMFDB = False;
field = {}
if F._data == None:
field['in_LMFDB'] = False
fld_coeffs = galmap['base_field']
pol = PolynomialRing(QQ, 'x')(fld_coeffs)
field['base_field'] = latex(pol)
field['isQQ'] = False
else:
field['in_LMFDB'] = True
if F.poly().degree() == 1:
field['isQQ'] = True
F.latex_poly = web_latex(F.poly())
field['base_field'] = F
galmapdatum = [
galmap['label'].split('-')[-1],
url_for_belyi_galmap_label(galmap['label']),
galmap['orbit_size'],
field,
galmap['triples_cyc'][0][0],
galmap['triples_cyc'][0][1],
galmap['triples_cyc'][0][2],
]
galmapdata.append(galmapdatum)
data['galmapdata'] = galmapdata
# Properties
properties = [('Label', passport['plabel']),
('Group', str(passport['group'])),
('Orders', str(passport['abc'])),
('Genus', str(passport['g'])),
('Size', str(passport['pass_size'])),
('Galois orbits', str(passport['num_orbits']))]
self.properties = properties
# Friends
self.friends = []
# Breadcrumbs
groupstr, abcstr, sigma0, sigma1, sigmaoo, gstr = data['plabel'].split(
"-")
lambdasstr = '%s-%s-%s' % (sigma0, sigma1, sigmaoo)
lambdasgstr = lambdasstr + "-" + gstr
self.bread = [('Belyi Maps', url_for(".index")),
(groupstr,
url_for(".by_url_belyi_search_group", group=groupstr)),
(abcstr,
url_for(".by_url_belyi_search_group_triple",
group=groupstr,
abc=abcstr)),
(lambdasgstr,
url_for(".by_url_belyi_passport_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g=gstr))]
# Title
self.title = "Passport " + data['plabel']
# Code snippets (only for curves)
self.code = {}
return
def lfuncEPhtml(L, fmt):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1, len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where, for " + pgoodset + ",\n"
if L.degree == 4 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for j in range(0, eulerlim):
this_prime = Primes().unrank(j)
if this_prime not in bad_primes:
good_primes.append(this_prime)
eptable = "<table id='eptable' class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th><th class='weight'>$F_p$</th>"
if L.degree > 2:
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
eptable += "</tr>\n"
eptable += "</thead>"
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
thispolygal = list_to_factored_poly_otherorder(lf[1], galois=True)
eptable += ("<tr><td>" + goodorbad + "</td><td>" + str(lf[0]) +
"</td><td>" + "$" + thispolygal[0] + "$" + "</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
if this_gal_group[0] == [0, 0]:
pass # do nothing, because the local faco is 1
elif this_gal_group[0] == [1, 1]:
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1],
'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
except IndexError:
eptable += "<tr><td></td><td>" + str(
j) + "</td><td>" + "not available" + "</td></tr>\n"
goodorbad = ""
goodorbad = "good"
firsttime = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)
eptable += ("<tr" + firsttime + "><td>" + goodorbad + "</td><td>" +
str(j) + "</td><td>" + "$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1) + "</td>"
# eptable += "</tr>\n"
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)
eptable += ("<tr" + firsttime + " class='more nodisplay'" + "><td>" +
goodorbad + "</td><td>" + str(j) + "</td><td>" + "$" +
list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)[0] +
"$" + "</td>")
if L.degree > 2:
this_gal_group = thispolygal[1]
eptable += "<td class='galois'>"
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
firsttime = ""
eptable += "<tr class='less toggle'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("more"); return true' + "'"
eptable += ' href="#moreep" '
eptable += ">show more</a></td></tr>\n"
eptable += "<tr class='more toggle nodisplay'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("less"); return true' + "'"
eptable += ' href="#eptable" '
eptable += ">show less</a></td></tr>\n"
eptable += "</table>\n"
ans += "\n" + eptable
return (ans)
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:
bread = get_bread([("Search Error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
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_display_short(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_display_knowl(gn, gt),
'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, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def pretty_galois_knowl(self):
return group_display_knowl(self._data['Galn'],self._data['Galt'])
def make_object(self, curve, endo, tama, ratpts, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = str(curve['Lhash'])
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex(factor(int(data['cond'])))
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1, 4, data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page",
cond=data['slabel'][0],
x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [(c[0],
list_to_factored_poly_otherorder(c[1]))
for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['abs_disc'])
data['disc'] = curve['disc_sign'] * data['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = list_to_min_eqn(data['min_eqn'])
data['disc_factor_latex'] = web_latex(factor(data['disc']))
data['igusa_clebsch'] = [
ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])
]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [
web_latex(zfactor(i)) for i in data['igusa_clebsch']
]
data['igusa_factor_latex'] = [
web_latex(zfactor(j)) for j in data['igusa']
]
data['aut_grp_id'] = curve['aut_grp_id']
data['geom_aut_grp_id'] = curve['geom_aut_grp_id']
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['has_square_sha'] = "square" if curve[
'has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",", 1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['torsion_order'] = curve['torsion_order']
data['torsion_factors'] = [
ZZ(a) for a in literal_eval(curve['torsion_subgroup'])
]
if len(data['torsion_factors']) == 0:
data['torsion_subgroup'] = '\mathrm{trivial}'
else:
data['torsion_subgroup'] = ' \\times '.join(
['\Z/{%s}\Z' % n for n in data['torsion_factors']])
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['tama'] = ''
for item in tama:
if item['tamagawa_number'] > 0:
tamgwnr = str(item['tamagawa_number'])
else:
tamgwnr = 'N/A'
data['tama'] += tamgwnr + ' (p = ' + str(item['p']) + '), '
data['tama'] = data['tama'][:-2] # trim last ", "
if ratpts:
if len(ratpts['rat_pts']):
data['rat_pts'] = ', '.join(
web_latex('(' + ' : '.join(map(str, P)) + ')')
for P in ratpts['rat_pts'])
data['rat_pts_v'] = 2 if ratpts['rat_pts_v'] else 1
# data['mw_rank'] = ratpts['mw_rank']
# data['mw_rank_v'] = ratpts['mw_rank_v']
else:
data['rat_pts_v'] = 0
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl(
curve['two_torsion_field'][0],
field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data[
'two_torsion_field_knowl'] = """splitting field of \(%s\) with Galois group %s""" % (
intlist_to_poly(
t[1]), group_display_knowl(t[2][0], t[2][1]))
else:
# invariants specific to isogeny class
curves_data = list(
db.g2c_curves.search({"class": curve['class']},
['label', 'eqn']))
if not curves_data:
raise KeyError(
"No curves found in database for isogeny class %s of genus 2 curve %s."
% (curve['class'], curve['label']))
data['curves'] = [{
"label":
c['label'],
"equation_formatted":
list_to_min_eqn(literal_eval(c['eqn'])),
"url":
url_for_curve_label(c['label'])
} for c in curves_data]
lfunc_data = db.lfunc_lfunctions.lucky(
{'Lhash': str(curve['Lhash'])})
if not lfunc_data:
raise KeyError(
"No Lfunction found in database for isogeny class of genus 2 curve %s."
% curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [
[nth_prime(n + 1), lfunc_data['euler_factors'][n]]
for n in range(len(lfunc_data['euler_factors']))
if nth_prime(n + 1) < 30 and (data['cond'] %
nth_prime(n + 1))
]
data['good_lfactors_pretty'] = [
(c[0], list_to_factored_poly_otherorder(c[1]))
for c in data['good_lfactors']
]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(
endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = """Endomorphism %s over \(\Q\):<br>""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_base'], endo['factorsRR_base'], ring=data['end_ring_base'] if is_curve else None)
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(
data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(
data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = """Endomorphism %s over \(\overline{\Q}\):""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_geom'], data['factorsRR_geom'], field=r'\overline{\Q}', ring=data['end_ring_geom'] if is_curve else None)
data['real_geom_end_alg_name'] = end_alg_name(
curve['real_geom_end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(
data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(
data['is_simple_geom'], data['split_field_label'],
data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(
endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'],
data.get('split_labels'),
data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
self.plot = encode_plot(
eqn_list_to_curve_plot(
data['min_eqn'],
data['rat_pts'].split(',') if 'rat_pts' in data else []))
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(
self.plot)
properties += [
(None, plot_link),
('Conductor', str(data['cond'])),
('Discriminant', str(data['disc'])),
]
properties += [
('Sato-Tate group', data['st_group_link']),
('\(\\End(J_{\\overline{\\Q}}) \\otimes \\R\)',
'\(%s\)' % data['real_geom_end_alg_name']),
('\(\\overline{\\Q}\)-simple',
bool_pretty(data['is_simple_geom'])),
('\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = [('L-function', data['lfunc_url'])]
if is_curve:
friends.append(('Isogeny class %s.%s' %
(data['slabel'][0], data['slabel'][1]),
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1])))
for friend_url in db.lfunc_instances.search({'Lhash': data['Lhash']},
'url'):
if '|' in friend_url:
for url in friend_url.split('|'):
add_friend(friends, lfunction_friend_from_url(url))
else:
add_friend(friends, lfunction_friend_from_url(friend_url))
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
add_friend(friends, ("Elliptic curve " + friend_label,
url_for_ec(friend_label)))
else:
add_friend(
friends,
("EC isogeny class " + ec_label_class(friend_label),
url_for_ec_class(friend_label)))
if is_curve:
friends.append(('Twists',
url_for(".index_Q",
g20=str(data['g2'][0]),
g21=str(data['g2'][1]),
g22=str(data['g2'][2]))))
# Breadcrumbs
self.bread = bread = [('Genus 2 Curves', url_for(".index")),
('$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0],
url_for(".by_conductor",
cond=data['slabel'][0])),
('%s' % data['slabel'][1],
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1]))]
if is_curve:
bread += [('%s' % data['slabel'][2],
url_for(".by_url_isogeny_class_discriminant",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2])),
('%s' % data['slabel'][3],
url_for(".by_url_curve_label",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2],
num=data['slabel'][3]))]
# Title
self.title = "Genus 2 " + ("Curve " if is_curve else
"Isogeny Class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage': '', 'magma': ''} # use default show names
code['curve'] = {
'sage':
'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s))'
% (data['min_eqn'][0], data['min_eqn'][1]),
'magma':
'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'
% (data['min_eqn'][0], data['min_eqn'][1])
}
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation(' + str(
data['cond']) + ',2),R!' + str(bad2) + '>*]'
else:
magma_cond_option = ''
code['cond'] = {
'magma':
'Conductor(LSeries(C%s)); Factorization($1);' % magma_cond_option
}
code['disc'] = {
'magma': 'Discriminant(C); Factorization(Integers()!$1);'
}
code['igusa_clebsch'] = {
'sage':
'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':
'IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];'
}
code['igusa'] = {
'magma':
'IgusaInvariants(C); [Factorization(Integers()!a): a in $1];'
}
code['g2'] = {'magma': 'G2Invariants(C);'}
code['aut'] = {'magma': 'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {
'magma':
'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'
}
code['num_rat_wpts'] = {
'magma': '#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'
}
if ratpts:
code['rat_pts'] = {
'magma':
'[' + ','.join([
"C![%s,%s,%s]" % (p[0], p[1], p[2])
for p in ratpts['rat_pts']
]) + '];'
}
code['two_selmer'] = {
'magma': 'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'
}
code['has_square_sha'] = {'magma': 'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {
'magma':
'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'
}
code['torsion_subgroup'] = {
'magma':
'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'
}
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\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:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
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()
ram_primes = D.prime_factors()
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'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf)
data['phrase'] = group_phrase(n, t, C)
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()[2]
rootofunity = Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a'))
else:
rootofunity = Ra('-1')
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': web_latex(rootofunity),
'fund_units': nf.units(),
'grh_label': grh_label
})
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 group', url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
('How data was computed', url_for(".how_computed_page"))]
if info['signature'] == [0,1]:
info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Label', label),
('Degree', '%s' % data['degree']),
('Signature', '$%s$' % data['signature']),
('Discriminant', '$%s$' % data['disc_factor']),
('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_display_short(data['degree'], t, C))
]
from lmfdb.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
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def pretty_galois_knowl(self):
return group_display_knowl(self._data['Galn'], self._data['Galt'])
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = lfdb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field ' + label
polynomial = coeff_to_poly(string2list(data['coeffs']))
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
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', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, db())),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = lfdb().find_one({'p': p, 'n': f, 'c': 0})
if unramdata is not None:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = lfdb().find_one({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
gsm = data['gsm']
if gsm == '0':
gsm = 'Not computed'
elif gsm == '-1':
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(string2list(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': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, db()),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], db()),
'unram': web_latex(unramp),
'eisen': web_latex(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, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def display_galois_group(self):
if self.galois_t == "": #the number field was not found in the database
return "The Galois group of this isogeny class is not in the database."
else:
C = getDBConnection()
return group_display_knowl(self.galois_n,self.galois_t,C)
def make_galmap_object(self, galmap):
from lmfdb.belyi.main import url_for_belyi_passport_label
# all information about the map goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
# the stuff that does not need to be polished
for elt in ('label', 'plabel', 'triples_cyc', 'orbit_size', 'g', 'abc',
'deg'):
data[elt] = galmap[elt]
nt = galmap['group'].split('T')
data['group'] = group_display_knowl(int(nt[0]), int(nt[1]),
getDBConnection())
data['geomtype'] = geomtypelet_to_geomtypename_dict[galmap['geomtype']]
data['lambdas'] = [str(c)[1:-1] for c in galmap['lambdas']]
data['isQQ'] = False
data['in_LMFDB'] = False
F = belyi_base_field(galmap)
if F._data == None:
fld_coeffs = galmap['base_field']
pol = PolynomialRing(QQ, 'x')(fld_coeffs)
data['base_field'] = latex(pol)
else:
data['in_LMFDB'] = True
if F.poly().degree() == 1:
data['isQQ'] = True
F.latex_poly = web_latex(F.poly())
data['base_field'] = F
crv_str = galmap['curve']
if crv_str == 'PP1':
data['curve'] = '\mathbb{P}^1'
else:
data['curve'] = make_curve_latex(crv_str)
# change pairs of floats to complex numbers
embeds = galmap['embeddings']
embed_strs = []
for el in embeds:
if el[1] < 0:
el_str = str(el[0]) + str(el[1]) + "\sqrt{-1}"
else:
el_str = str(el[0]) + "+" + str(el[1]) + "\sqrt{-1}"
embed_strs.append(el_str)
data['map'] = make_map_latex(galmap['map'])
data['embeddings_and_triples'] = []
if data['isQQ']:
for i in range(0, len(data['triples_cyc'])):
triple_cyc = data['triples_cyc'][i]
data['embeddings_and_triples'].append([
"\\text{not applicable (over $\mathbb{Q}$)}",
triple_cyc[0], triple_cyc[1], triple_cyc[2]
])
else:
for i in range(0, len(data['triples_cyc'])):
triple_cyc = data['triples_cyc'][i]
data['embeddings_and_triples'].append([
embed_strs[i], triple_cyc[0], triple_cyc[1], triple_cyc[2]
])
data['lambdas'] = [str(c)[1:-1] for c in galmap['lambdas']]
# Properties
properties = [
('Label', galmap['label']),
('Group', str(galmap['group'])),
('Orders', str(galmap['abc'])),
('Genus', str(galmap['g'])),
('Size', str(galmap['orbit_size'])),
]
self.properties = properties
# Friends
self.friends = [('Passport',
url_for_belyi_passport_label(galmap['plabel']))]
# Breadcrumbs
groupstr, abcstr, sigma0, sigma1, sigmaoo, gstr, letnum = data[
'label'].split("-")
lambdasstr = '%s-%s-%s' % (sigma0, sigma1, sigmaoo)
lambdasgstr = lambdasstr + "-" + gstr
self.bread = [
('Belyi Maps', url_for(".index")),
(groupstr, url_for(".by_url_belyi_search_group", group=groupstr)),
(abcstr,
url_for(".by_url_belyi_search_group_triple",
group=groupstr,
abc=abcstr)),
(lambdasgstr,
url_for(".by_url_belyi_passport_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g=gstr)),
(letnum,
url_for(".by_url_belyi_galmap_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g=gstr,
letnum=letnum)),
]
# Title
self.title = "Belyi map " + data['label']
# Code snippets (only for curves)
self.code = {}
return
def display_galois_group(self):
if not hasattr(self, 'galois_t') or not self.galois_t: #the number field was not found in the database
return "The Galois group of this isogeny class is not in the database."
else:
return group_display_knowl(self.galois_n, self.galois_t)
def pretty_galois_knowl(self):
C = getDBConnection()
return group_display_knowl(self._data['Galois_nt'][0],
self._data['Galois_nt'][1], C)
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label})
bread = get_bread([(label, ' ')])
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.render_artin_representation_webpage', label=clean_label), 301)
try:
the_rep = ArtinRepresentation(label)
except:
flash(Markup("Error: <span style='color:black'>%s</span> is not the label of an Artin representation in the database." % (label)), "error")
return search_input_error({'err':''}, bread)
extra_data = {} # for testing?
extra_data['galois_knowl'] = group_display_knowl(5,3) # for testing?
#artin_logger.info("Found %s" % (the_rep._data))
title = "Artin Representation %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [("Label", label),
("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
#("Conductor", str(the_rep.conductor())),
("Conductor", "$" + the_rep.factored_conductor_latex() + "$"),
#("Bad primes", str(the_rep.bad_primes())),
("Root number", processed_root_number),
("Frobenius-Schur indicator", str(the_rep.indicator()))
]
friends = []
nf_url = the_nf.url_for()
if nf_url:
friends.append(("Artin Field", nf_url))
cc = the_rep.central_character()
if cc is not None:
if the_rep.dimension()==1:
if cc.order == 2:
cc_name = cc.symbol
else:
cc_name = cc.texname
friends.append(("Dirichlet character "+cc_name, url_for("characters.render_Dirichletwebpage", modulus=cc.modulus, number=cc.number)))
else:
detrep = the_rep.central_character_as_artin_rep()
friends.append(("Determinant representation "+detrep.label(), detrep.url_for()))
# once the L-functions are in the database, the link can always be shown
#if the_rep.dimension() <= 6:
if the_rep.dimension() == 1:
# Zeta is loaded differently
if cc.modulus == 1 and cc.number == 1:
friends.append(("L-function", url_for("l_functions.l_function_dirichlet_page", modulus=cc.modulus, number=cc.number)))
else:
# looking for Lhash dirichlet_L_modulus.number
mylhash = 'dirichlet_L_%d.%d'%(cc.modulus,cc.number)
lres = db.lfunc_instances.lucky({'Lhash': mylhash})
if lres is not None:
friends.append(("L-function", url_for("l_functions.l_function_dirichlet_page", modulus=cc.modulus, number=cc.number)))
# Dimension > 1
elif int(the_rep.conductor())**the_rep.dimension() <= 729000000000000:
friends.append(("L-function", url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
info={}
#mychar = the_rep.central_char()
#info['pol2']= str([((j+1),mychar(j+1, 2*the_rep.character_field())) for j in range(50)])
#info['pol3']=str(the_rep.central_character())
#info['pol3']=str(the_rep.central_char(3))
#info['pol5']=str(the_rep.central_char(5))
#info['pol7']=str(the_rep.central_char(7))
#info['pol11']=str(the_rep.central_char(11))
learnmore=[('Artin representations labels', url_for(".labels_page"))]
return render_template("artin-representation-show.html", credit=tim_credit, support=support_credit, title=title, bread=bread, friends=friends, object=the_rep, properties2=properties, extra_data=extra_data, info=info, learnmore=learnmore)
def lfuncEPhtml(L,fmt):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1,len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where, for " + pgoodset + ",\n"
if L.degree == 4 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for j in range(0, eulerlim):
this_prime = Primes().unrank(j)
if this_prime not in bad_primes:
good_primes.append(this_prime)
eptable = "<table id='eptable' class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th><th class='weight'>$F_p$</th>"
if L.degree > 2:
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
eptable += "</tr>\n"
eptable += "</thead>"
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
thispolygal = list_to_factored_poly_otherorder(lf[1], galois=True)
eptable += ("<tr><td>" + goodorbad + "</td><td>" + str(lf[0]) + "</td><td>" +
"$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
if this_gal_group[0]==[0,0]:
pass # do nothing, because the local faco is 1
elif this_gal_group[0]==[1,1]:
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1],'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
except IndexError:
eptable += "<tr><td></td><td>" + str(j) + "</td><td>" + "not available" + "</td></tr>\n"
goodorbad = ""
goodorbad = "good"
firsttime = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(L.localfactors[this_prime_index],galois=True)
eptable += ("<tr" + firsttime + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1) + "</td>"
# eptable += "</tr>\n"
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(L.localfactors[this_prime_index],galois=True)
eptable += ("<tr" + firsttime + " class='more nodisplay'" + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + list_to_factored_poly_otherorder(L.localfactors[this_prime_index], galois=True)[0] + "$" +
"</td>")
if L.degree > 2:
this_gal_group = thispolygal[1]
eptable += "<td class='galois'>"
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
firsttime = ""
eptable += "<tr class='less toggle'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("more"); return true' + "'"
eptable += ' href="#moreep" '
eptable += ">show more</a></td></tr>\n"
eptable += "<tr class='more toggle nodisplay'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("less"); return true' + "'"
eptable += ' href="#eptable" '
eptable += ">show less</a></td></tr>\n"
eptable += "</table>\n"
ans += "\n" + eptable
return(ans)
def render_field_webpage(args):
data = None
info = {}
if "label" in args:
label = clean_input(args["label"])
C = base.getDBConnection()
data = C.localfields.fields.find_one({"label": label})
if data is None:
bread = get_bread([("Search error", " ")])
info["err"] = "Field " + label + " was not found in the database."
info["label"] = label
return search_input_error(info, bread)
title = "Local Number Field " + label
polynomial = coeff_to_poly(data["coeffs"])
p = data["p"]
e = data["e"]
f = data["f"]
cc = data["c"]
GG = data["gal"]
gn = GG[0]
gt = GG[1]
prop2 = [
("Label", label),
("Base", "\(\Q_{%s}\)" % p),
("Degree", "\(%s\)" % data["n"]),
("e", "\(%s\)" % e),
("f", "\(%s\)" % f),
("c", "\(%s\)" % cc),
("Galois group", group_display_short(gn, gt, C)),
]
Pt = PolynomialRing(QQ, "t")
Pyt = PolynomialRing(Pt, "y")
eisenp = Pyt(str(data["eisen"]))
unramp = Pyt(str(data["unram"]))
# Look up the unram poly so we can link to it
unramdata = C.localfields.fields.find_one({"p": p, "n": f, "c": 0})
if len(unramdata) > 0:
unramfriend = "/LocalNumberField/%s" % unramdata["label"]
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ""
rfdata = C.localfields.fields.find_one({"p": p, "n": {"$in": [1, 2]}, "rf": data["rf"]})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ""
else:
rffriend = "/LocalNumberField/%s" % rfdata["label"]
info.update(
{
"polynomial": web_latex(polynomial),
"n": data["n"],
"p": data["p"],
"c": data["c"],
"e": data["e"],
"f": data["f"],
"t": data["t"],
"u": data["u"],
"rf": printquad(data["rf"], p),
"hw": data["hw"],
"slopes": show_slopes(data["slopes"]),
"gal": group_display_knowl(gn, gt, C),
"gt": gt,
"inertia": group_display_inertia(data["inertia"], C),
"unram": web_latex(unramp),
"eisen": web_latex(eisenp),
"gms": data["gms"],
"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, " ")])
learnmore = [
("Completeness of the data", url_for(".completeness_page")),
("Source of the data", url_for(".how_computed_page")),
("Local field labels", url_for(".labels_page")),
]
return render_template(
"lf-show-field.html",
credit=LF_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore,
)
