def __init__(self, fund_group):
self.domain_gens = fund_group.generators()
ab_words = [
abelianize_word(R, self.domain_gens)
for R in fund_group.relators()
]
R = matrix(ZZ, ab_words).transpose()
D, U, V = R.smith_form()
m = U.nrows()
assert m == D.nrows()
d = min(D.nrows(), D.ncols())
diag = D.diagonal()
num_ones = diag.count(1)
self.elementary_divisors = diag[num_ones:] + [
0,
] * (m - d)
self.U = U[num_ones:]
tor = [d for d in self.elementary_divisors if d != 0]
free = [d for d in self.elementary_divisors if d == 0]
names = []
if len(tor) == 1:
names.append('u')
else:
names += ['u%d' % i for i in range(len(tor))]
if len(free) == 1:
names.append('t')
else:
names += ['t%d' % i for i in range(len(free))]
self._range = AbelianGroup(self.elementary_divisors, names=names)
def generators_of_subgroups_of_unit_group(R):
"""
INPUT:
- R - a commutative ring whose unit group is finite
OUTPUT:
- An iterator which yields a set of generators for each subgroup of R^*
EXAMPLES::
sage: from mdsage import *
sage: list(generators_of_subgroups_of_unit_group(Integers(28)))
[[15, 17], [11], [3], [13, 15], [15], [27], [17], [9], [13], []]
"""
gens = R.unit_gens()
invariants = [g.multiplicative_order() for g in gens]
assert all(i!=0 for i in invariants)
A=AbelianGroup(invariants)
for G in A.subgroups():
yield [prod(f**e for f,e in zip(gens,g.exponents())) for g in G.gens()]
def generators_of_subgroups_of_unit_group(R):
"""
INPUT:
- R - a commutative ring whose unit group is finite
OUTPUT:
- An iterator which yields a set of generators for each subgroup of R^*
EXAMPLES::
sage: from mdsage import *
sage: list(generators_of_subgroups_of_unit_group(Integers(28)))
[[15, 17], [11], [3], [13, 15], [15], [27], [17], [9], [13], []]
"""
gens = R.unit_gens()
invariants = [g.multiplicative_order() for g in gens]
assert all(i!=0 for i in invariants)
A=AbelianGroup(invariants)
for G in A.subgroups():
yield [prod(f**e for f,e in zip(gens,g.exponents())) for g in G.gens()]
def render_field_webpage(args):
data = None
if 'label' in args:
label = str(args['label'])
import base
C = base.getDBConnection()
data = C.numberfields.fields.find_one({'label': label})
if data is None:
return "No such field: " + label + " in the database"
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
K = coeff_to_nf(data['coefficients'])
D = data['discriminant']
h = data['class_number']
data['galois_group'] = str(data['galois_group'][3])
data['class_group_invs'] = data['class_group']
if data['class_group_invs'] == []:
data['class_group_invs'] = 'Trivial'
data['class_group'] = str(AbelianGroup(data['class_group']))
sig = data['signature']
D = ZZ(data['discriminant'])
ram_primes = D.prime_factors()
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
Gorder, Gsign, Gab = GG_data(data['galois_group'])
if Gab:
Gab = 'abelian'
else:
Gab = 'non-abelian'
unit_rank = sig[0] + sig[1] - 1
if unit_rank == 0:
reg = 1
else:
reg = K.regulator()
UK = K.unit_group()
info.update(data)
info.update({
'label':
field_pretty(label),
'polynomial':
web_latex(K.defining_polynomial()),
'ram_primes':
ram_primes,
'integral_basis':
web_latex(K.integral_basis()),
'regulator':
web_latex(reg),
'unit_rank':
unit_rank,
'root_of_unity':
web_latex(UK.torsion_generator()),
'fund_units':
', '.join([web_latex(u) for u in UK.fundamental_units()]),
'Gorder':
Gorder,
'Gsign':
Gsign,
'Gab':
Gab
})
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
# info['friends'] = [('L-function', '/')]
info['friends'] = [('L-function', "/L/NumberField/%s" % label)]
info['learnmore'] = [
('Number Field labels', url_for("render_labels_page")),
('Galois group labels', url_for("render_groups_page")),
('Discriminant ranges', url_for("render_discriminants_page"))
]
bread = [('Number Fields', url_for("number_field_render_webpage")),
('%s' % info['label'], ' ')]
t = "Number Field %s" % info['label']
properties = ['<br>']
properties.extend('<table>')
properties.extend(
'<tr><td align=left><b>Degree:</b><td align=left> %s</td>' %
data['degree'])
properties.extend(
'<tr><td align=left><b>Signature:</b><td align=left>%s</td>' %
data['signature'])
properties.extend(
'<tr><td align=left><b>Discriminant:</b><td align=left>%s</td>' %
data['discriminant'])
if npr == 1:
properties.extend(
'<tr><td align=left><b>Ramified prime:</b><td align=left>%s</td>' %
ram_primes)
else:
if npr == 0:
properties.extend(
'<tr><td align=left><b>Ramified primes:</b><td align=left>%s</td>'
% "None")
else:
properties.extend(
'<tr><td align=left><b>Ramified primes:</b><td align=left>%s</td>'
% ram_primes)
properties.extend(
'<tr><td align=left><b>Class number:</b><td align=left>%s</td>' %
data['class_number'])
properties.extend(
'<tr><td align=left><b>Class group:</b><td align=left>%s</td>' %
data['class_group_invs'])
properties.extend(
'<tr><td align=left><b>Galois group:</b><td align=left>%s</td>' %
data['galois_group'])
properties.extend('</table>')
del info['_id']
return render_template("number_field/number_field.html",
properties=properties,
credit=NF_credit,
title=t,
bread=bread,
friends=info.pop('friends'),
info=info)
def class_group(self):
if self.haskey('class_group'):
cg_list = self._data['class_group']
return str(AbelianGroup(
cg_list)) + ', order ' + self.class_number_latex()
return na_text()
def class_group(self):
if self.haskey('class_group'):
cg_list = string2list(self._data['class_group'])
return str(AbelianGroup(cg_list)) + ', order ' + str(self._data['class_number'])
return na_text()