def render_field_webpage(args):
data = None
info = {}
bread = bread_prefix()
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
flash_error("Number field %s was not found in the database.",
label)
else:
flash_error("%s is not a valid label for a number field.", label)
return redirect(url_for(".number_field_render_webpage"))
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number_latex()
ram_primes = nf.ramified_primes()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['autstring'] = r'\Gal' if data['is_galois'] else r'\Aut'
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if dirichlet_chars:
data['dirichlet_group'] = [
r'<a href = "%s">$\chi_{%s}(%s,·)$</a>' %
(url_for('characters.render_Dirichletwebpage',
modulus=data['conductor'],
number=j), data['conductor'], j)
for j in dirichlet_chars
]
if len(data['dirichlet_group']) == 1:
data[
'dirichlet_group'] = r'<span style="white-space:nowrap">$\lbrace$' + data[
'dirichlet_group'][0] + r'$\rbrace$</span>'
else:
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(
data['dirichlet_group']
[:-1]) + '<span style="white-space:nowrap">' + data[
'dirichlet_group'][-1] + r'$\rbrace$</span>'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = r"\(%s\)" % str(data['conductor'])
else:
factored_conductor = factor_base_factor(data['conductor'],
ram_primes)
factored_conductor = factor_base_factorization_latex(
factored_conductor)
data['conductor'] = r"\(%s=%s\)" % (str(
data['conductor']), factored_conductor)
data['galois_group'] = group_pretty_and_nTj(n, t, True)
data['auts'] = db.gps_transitive.lookup(r'{}T{}'.format(n, t))['auts']
data['cclasses'] = cclasses_display_knowl(n, t)
data['character_table'] = character_table_display_knowl(n, t)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = bigint_knowl(D, cutoff=60, sides=3)
else:
data['discriminant'] = bigint_knowl(
D, cutoff=60,
sides=3) + r"\(\medspace = %s\)" % data['disc_factor']
if nf.frobs():
data['frob_data'], data['seeram'] = see_frobs(nf.frobs())
else: # fallback in case we haven't computed them in a case
data['frob_data'], data['seeram'] = frobs(nf)
# This could put commas in the rd, we don't want to trigger spaces
data['rd'] = ('$%s$' % fixed_prec(nf.rd(), 2)).replace(',', '{,}')
# Bad prime information
npr = len(ram_primes)
ramified_algebras_data = nf.ramified_algebras_data()
if isinstance(ramified_algebras_data, str):
loc_alg = ''
else:
# [label, latex, e, f, c, gal]
loc_alg = ''
for j in range(npr):
if ramified_algebras_data[j] is None:
loc_alg += '<tr><td>%s<td colspan="7">Data not computed' % str(
ram_primes[j]).rstrip('L')
else:
from lmfdb.local_fields.main import show_slope_content
mydat = ramified_algebras_data[j]
p = ram_primes[j]
loc_alg += '<tr><td rowspan="%d">$%s$</td>' % (len(mydat),
str(p))
mm = mydat[0]
myurl = url_for('local_fields.by_label', label=mm[0])
lab = mm[0]
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
for mm in mydat[1:]:
lab = mm[0]
myurl = url_for('local_fields.by_label', label=lab)
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
loc_alg += '</tbody></table>'
ram_primes = str(ram_primes)[1:-1]
# Get rid of python L for big numbers
ram_primes = ram_primes.replace('L', '')
if not ram_primes:
ram_primes = r'\textrm{None}'
data['phrase'] = group_phrase(n, t)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh(
) else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'computed' in str(data['class_number']):
grh_lab = ''
grh_label = ''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
rootofunity = '%s (order $%d$)' % (nf.root_of_1_gen(),
nf.root_of_1_order())
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex(nf.poly()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': rootofunity,
'fund_units': nf.units_safe(),
'cnf': nf.cnf(),
'grh_label': grh_label,
'loc_alg': loc_alg
})
bread.append(('%s' % nf_label_pretty(info['label_raw']), ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet character group',
url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
str(a) for a in dirichlet_chars),
poly=info['polynomial'])))
resinfo = []
galois_closure = nf.galois_closure()
if galois_closure[0] > 0:
if galois_closure[1]:
resinfo.append(('gc', galois_closure[1]))
if galois_closure[2]:
info['friends'].append(('Galois closure',
url_for(".by_label",
label=galois_closure[2][0])))
else:
resinfo.append(('gc', [dnc]))
sextic_twins = nf.sextic_twin()
if sextic_twins[0] > 0:
if sextic_twins[1]:
resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1])))
else:
resinfo.append(('sex', dnc))
siblings = nf.siblings()
# [degsib list, label list]
# first is list of [deg, num expected, list of knowls]
if siblings[0]:
for sibdeg in siblings[0]:
if not sibdeg[2]:
sibdeg[2] = dnc
else:
nsibs = len(sibdeg[2])
sibdeg[2] = ', '.join(sibdeg[2])
if nsibs < sibdeg[1]:
sibdeg[2] += ', some ' + dnc
resinfo.append(('sib', siblings[0]))
for lab in siblings[1]:
if lab:
labparts = lab.split('.')
info['friends'].append(("Degree %s sibling" % labparts[0],
url_for(".by_label", label=lab)))
arith_equiv = nf.arith_equiv()
if arith_equiv[0] > 0:
if arith_equiv[1]:
resinfo.append(
('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1])))
for aelab in arith_equiv[2]:
info['friends'].append(('Arithmetically equivalent sibling',
url_for(".by_label", label=aelab)))
else:
resinfo.append(('ae', dnc, len(arith_equiv[1])))
info['resinfo'] = resinfo
learnmore = learnmore_list()
title = "Number field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
if len(ram_primes) > 30:
ram_primes = 'see page'
else:
ram_primes = '$%s$' % ram_primes
properties = [('Label', nf_label_pretty(label)),
('Degree', prop_int_pretty(data['degree'])),
('Signature', '$%s$' % data['signature']),
('Discriminant', prop_int_pretty(D)),
('Root discriminant', '%s' % data['rd']),
('Ramified ' + primes + '', ram_primes),
('Class number', '%s %s' % (data['class_number'], grh_lab)),
('Class group',
'%s %s' % (data['class_group_invs'], grh_lab)),
('Galois group', group_pretty_and_nTj(data['degree'], t))]
downloads = [('Stored data to gp',
url_for('.nf_download', nf=label, download_type='data'))]
for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]:
downloads.append(('Download {} code'.format(lang[0]),
url_for(".nf_download",
nf=label,
download_type=lang[1])))
from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
from lmfdb.artin_representations.math_classes import artin_label_pretty
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
arts = [
z.label()
for z in info["tim_number_field"].artin_representations()
]
#print arts
for ar in arts:
info['friends'].append((
'Artin representation ' + artin_label_pretty(ar),
url_for(
"artin_representations.render_artin_representation_webpage",
label=ar)))
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m == 0:
return ''
if m == 1:
return '*'
return '*<sup>%d</sup>' % m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
return render_template("nf-show-field.html",
properties=properties,
credit=NF_credit,
title=title,
bread=bread,
code=nf.code,
friends=info.pop('friends'),
downloads=downloads,
learnmore=learnmore,
info=info,
KNOWL_ID="nf.%s" % label)
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label, 'search_array': ArtinSearchArray()})
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.render_artin_representation_webpage', label=clean_label), 301)
# We could have a single representation or a Galois orbit
case = parse_any(label)
if case[0] == 'malformed':
try:
raise ValueError
except Exception:
flash_error("%s is not in a valid form for the label for an Artin representation or a Galois orbit of Artin representations", label)
return redirect(url_for(".index"))
# Do this twice to customize error messages
newlabel = case[1]
case = case[0]
if case == 'rep':
try:
the_rep = ArtinRepresentation(newlabel)
except Exception:
newlabel = parse_artin_label(label)
flash_error("Artin representation %s is not in database", label)
return redirect(url_for(".index"))
else: # it is an orbit
try:
the_rep = ArtinRepresentation(newlabel+'.a')
except Exception:
newlabel = parse_artin_orbit_label(newlabel)
flash_error("Galois orbit of Artin representations %s is not in database", label)
return redirect(url_for(".index"))
# in this case we want all characters
num_conj = the_rep.galois_conjugacy_size()
allchars = [ ArtinRepresentation(newlabel+'.'+num2letters(j)).character_formatted() for j in range(1,num_conj+1)]
label = newlabel
bread = get_bread([(artin_label_pretty(label), ' ')])
#artin_logger.info("Found %s" % (the_rep._data))
if case=='rep':
title = "Artin representation %s" % label
else:
title = "Galois orbit of Artin representations %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = '$%s$' % the_rep.sign()
properties = [("Label", artin_label_pretty(label)),
("Dimension", prop_int_pretty(the_rep.dimension())),
("Group", the_rep.group()),
("Conductor", prop_int_pretty(the_rep.conductor()))]
if case == 'rep':
properties.append( ("Root number", processed_root_number) )
properties.append( ("Indicator", prop_int_pretty(the_rep.indicator())) )
friends = []
wnf = None
proj_wnf = None
nf_url = the_nf.url_for()
if nf_url:
friends.append(("Field {}".format(the_nf.label()), nf_url))
wnf = the_nf.wnf()
proj_wnf = WebNumberField.from_coeffs(the_rep._data['Proj_Polynomial'])
if proj_wnf.is_in_db():
proj_coefs = [int(z) for z in proj_wnf.coeffs()]
if proj_coefs != the_nf.polynomial():
friends.append(("Field {}".format(proj_wnf.get_label()),
str(url_for("number_fields.by_label", label=proj_wnf.get_label()))))
if case == 'rep':
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 "+detrep.label(), detrep.url_for()))
add_lfunction_friends(friends,label)
# 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())))
orblabel = re.sub(r'\.[a-z]+$', '', label)
friends.append(("Galois orbit " + artin_label_pretty(orblabel),
url_for(".render_artin_representation_webpage", label=orblabel)))
else:
add_lfunction_friends(friends,label)
friends.append(("L-function", url_for("l_functions.l_function_artin_page", label=the_rep.label())))
for j in range(1,1+the_rep.galois_conjugacy_size()):
newlabel = label+'.'+num2letters(j)
friends.append(("Artin representation " + artin_label_pretty(newlabel),
url_for(".render_artin_representation_webpage", label=newlabel)))
info={} # for testing
if case == 'rep':
return render_template(
"artin-representation-show.html",
title=title,
bread=bread,
friends=friends,
object=the_rep,
cycle_string=cycle_string,
wnf=wnf,
proj_wnf=proj_wnf,
properties=properties,
info=info,
learnmore=learnmore_list(),
KNOWL_ID="artin.%s" % label,
)
# else we have an orbit
return render_template(
"artin-representation-galois-orbit.html",
title=title,
bread=bread,
allchars=allchars,
friends=friends,
object=the_rep,
cycle_string=cycle_string,
wnf=wnf,
proj_wnf=proj_wnf,
properties=properties,
info=info,
learnmore=learnmore_list(),
KNOWL_ID="artin.%s" % label,
)
