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' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
from lmfdb.WebNumberField import nf_display_knowl
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.K())
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)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'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 = [('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, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
if 'label' in args:
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
label2 = re.sub(r'[<>]', '', args['label'])
if 'You need to enter a field' in label2:
info['err'] = label2
else:
info['err'] = 'No such field: %s in the database' % label2
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()
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for("render_Character", arg1=data['conductor'], arg2=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()
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.K())
data['phrase'] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst('a')
zk = list(zk.nf_get_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()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'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():
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("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"))]
# 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 = [('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.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ['*' if x>0 else '' 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, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)