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