def field_label(F, pretty=True, check=False):
r"""
Returns the LMFDB label of the field F.
"""
if F.absolute_degree() == 1:
p = 'x'
else:
pp = F.absolute_polynomial()
x = pp.parent().gen()
p = str(pp).replace(str(x), 'x')
l = poly_to_field_label(p)
if l is None:
if check:
return False
else:
if pretty:
return web_latex_split_on_pm(pp)
else:
return pp
else:
if check:
return True
if pretty:
return field_pretty(l)
else:
return l
def find_field(pol, verbose=False):
"""
pol is a string holding a list of coefficients, constant first, 1 last, e.g. '-2,0,1'
Looks up this defining polynomial kn LMFDB and returns its label, or None
"""
coeffs = str_to_list(pol)
deg = len(coeffs)-1
if deg==2:
c, b, a = coeffs
d = ZZ(b*b-4*a*c).squarefree_part()
D = d if (d-1)%4==0 else 4*d
absD = D.abs()
s = 0 if d<0 else 2
return '2.{}.{}.1'.format(s,absD)
from lmfdb.number_fields.number_field import poly_to_field_label
poly = Qx(coeffs)
Flabel = poly_to_field_label(poly)
if Flabel==None:
print("********* field with polynomial {} is not in the database!".format(poly))
K = NumberField(poly, 'a')
poly = K.optimized_representation()[0].defining_polynomial()
print("********* using optimised polynomial {}".format(poly))
return poly_to_str(poly)
else:
if verbose:
print("{} has label {}".format(pol,Flabel))
return Flabel
def field_label(F, pretty = True, check=False):
r"""
Returns the LMFDB label of the field F.
"""
if F.absolute_degree() == 1:
p = 'x'
else:
pp = F.absolute_polynomial()
x = pp.parent().gen()
p = str(pp).replace(str(x), 'x')
l = poly_to_field_label(p)
if l is None:
if check:
return False
else:
if pretty:
return web_latex_split_on_pm(pp)
else:
return pp
else:
if check:
return True
if pretty:
return field_pretty(l)
else:
return l
def find_field(pol, verbose=False):
"""
pol is a string holding a list of coefficients, constant first, 1 last, e.g. '-2,0,1'
Looks up this defining polynomial kn LMFDB and returns its label, or None
"""
coeffs = str_to_list(pol)
deg = len(coeffs) - 1
if deg == 2:
c, b, a = coeffs
d = ZZ(b * b - 4 * a * c).squarefree_part()
D = d if (d - 1) % 4 == 0 else 4 * d
absD = D.abs()
s = 0 if d < 0 else 2
return '2.{}.{}.1'.format(s, absD)
from lmfdb.number_fields.number_field import poly_to_field_label
poly = Qx(coeffs)
Flabel = poly_to_field_label(poly)
if Flabel is None:
print("********* field with polynomial {} is not in the database!".
format(poly))
K = NumberField(poly, 'a')
poly = K.optimized_representation()[0].defining_polynomial()
print("********* using optimised polynomial {}".format(poly))
return poly_to_str(poly)
else:
if verbose:
print("{} has label {}".format(pol, Flabel))
return Flabel
def to_db(self):
r"""
We store the LMFDB label of the absolute field
in the db.
"""
K = self._value
if K.absolute_degree() == 1:
p = 'x'
else:
p = K.absolute_polynomial()
l = poly_to_field_label(p)
return l
def to_db(self):
r"""
We store the LMFDB label of the absolute field
in the db.
"""
K = self._value
if K.absolute_degree() == 1:
p = 'x'
else:
p = K.absolute_polynomial()
l = poly_to_field_label(p)
return l
def add_lattice_nf(ll):
n_field, gram_input = ll
gram_input = [[int(i) for i in l] for l in gram_input]
R = PolynomialRing(QQ, 'x')
nf_label = poly_to_field_label(R(n_field))
lattice = l1.find_one({'gram': gram_input})
if lattice is None:
n = len(gram_input[0])
d = matrix(gram_input).determinant()
result = [
B for B in l1.find({
'dim': int(n),
'det': int(d)
}) if isom(gram_input, B['gram'])
]
if len(result) == 1:
lat_label = result[0]['label']
is_lat_in = "yes"
elif len(result) > 1:
print "... need to be checked ..."
print "***********"
else:
lat_label = "new"
is_lat_in = gram_input
else:
lat_label = lattice['label']
is_lat_in = "yes"
try:
lab = nf_label + lat_label
except:
print nf_label, lat_label
print "fail"
res = l2.find_one({'label': lab})
if res is None:
print "new data"
if saving:
l2.insert_one({
'nf_label': nf_label,
'lat_label': lat_label,
'is_lat_in': is_lat_in,
'label': lab
})
else:
print "data already in the database"
def to_db(self):
r"""
We store the LMFDB label of the absolute field in the db.
"""
if self._db_value_has_been_set and not self._db_value is None:
return self._db_value
K = self._value
if hasattr(K, "lmfdb_label"):
return K.lmfdb_label
if K.absolute_degree() == 1:
p = 'x'
else:
p = K.absolute_polynomial()
l = poly_to_field_label(p)
return l
def to_db(self):
r"""
We store the LMFDB label of the absolute field in the db.
"""
if self._db_value_has_been_set and not self._db_value is None:
return self._db_value
K = self._value
if hasattr(K, "lmfdb_label"):
return K.lmfdb_label
if K.absolute_degree() == 1:
p = 'x'
else:
p = K.absolute_polynomial()
l = poly_to_field_label(p)
return l
def add_lattice_nf(ll):
n_field,gram_input = ll
gram_input=[[int(i) for i in l] for l in gram_input]
R = PolynomialRing(QQ, 'x');
nf_label = poly_to_field_label(R(n_field))
lattice = l1.find_one({'gram': gram_input })
if lattice is None:
n=len(gram_input[0])
d=matrix(gram_input).determinant()
result=[B for B in l1.find({'dim': int(n), 'det' : int(d)}) if isom(gram_input, B['gram'])]
if len(result)==1:
lat_label =result[0]['label']
is_lat_in = "yes"
elif len(result)>1:
print "... need to be checked ..."
print "***********"
else :
lat_label = "new"
is_lat_in = gram_input
else:
lat_label=lattice['label']
is_lat_in = "yes"
try:
lab=nf_label+lat_label
except:
print nf_label, lat_label
print "fail"
res=l2.find_one({'label': lab })
if res is None:
print "new data"
if saving:
l2.insert_one({'nf_label': nf_label, 'lat_label': lat_label, 'is_lat_in' : is_lat_in, 'label': lab})
else:
print "data already in the database"
def render_sample_page(family, sam, args, bread):
info = {
'args': to_dict(args),
'sam': sam,
'latex': latex,
'type': sam.type(),
'name': sam.name(),
'full_name': sam.full_name(),
'weight': sam.weight(),
'fdeg': sam.degree_of_field(),
'is_eigenform': sam.is_eigenform(),
'field_poly': sam.field_poly()
}
if sam.is_integral() != None:
info['is_integral'] = sam.is_integral()
if 'Sp4Z' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=0)
if 'Sp4Z_2' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=2)
info['space'] = '$' + family.latex_name.replace(
'k', '{' + str(sam.weight()) + '}') + '$'
if 'space_url' in info:
bread.append((info['space'], info['space_url']))
info['space_href'] = '<a href="%s">%s</d>' % (
info['space_url'],
info['space']) if 'space_url' in info else info['space']
if info['field_poly'].disc() < 10**10:
label = poly_to_field_label(info['field_poly'])
if label:
info['field_label'] = label
info['field_url'] = url_for('number_fields.by_label', label=label)
info['field_href'] = '<a href="%s">%s</a>' % (info['field_url'],
field_pretty(label))
bread.append((info['name'], ''))
title = 'Siegel modular forms sample ' + info['full_name']
properties = [('Space', info['space_href']), ('Name', info['name']),
('Type', '<br>'.join(info['type'].split(','))),
('Weight', str(info['weight'])),
('Hecke eigenform', str(info['is_eigenform'])),
('Field degree', str(info['fdeg']))]
try:
evs_to_show = parse_ints_to_list_flash(args.get('ev_index'),
'list of $l$')
fcs_to_show = parse_ints_to_list_flash(args.get('fc_det'),
'list of $\\det(F)$')
except ValueError:
evs_to_show = []
fcs_to_show = []
info['evs_to_show'] = sorted([
n for n in
(evs_to_show if len(evs_to_show) else sam.available_eigenvalues()[:10])
])
info['fcs_to_show'] = sorted([
n for n in (fcs_to_show if len(fcs_to_show) else sam.
available_Fourier_coefficients()[1:6])
])
info['evs_avail'] = [n for n in sam.available_eigenvalues()]
info['fcs_avail'] = [n for n in sam.available_Fourier_coefficients()]
# Do not attempt to constuct a modulus ideal unless the field has a reasonably small discriminant
# otherwise sage may not even be able to factor the discriminant
info['field'] = sam.field()
if info['field_poly'].disc() < 10**80:
null_ideal = sam.field().ring_of_integers().ideal(0)
info['modulus'] = null_ideal
modulus = args.get('modulus', '').strip()
m = 0
if modulus:
try:
O = sam.field().ring_of_integers()
m = O.ideal([O(str(b)) for b in modulus.split(',')])
except Exception:
info['error'] = True
flash_error(
"Unable to construct modulus ideal from specified generators %s.",
modulus)
if m == 1:
info['error'] = True
flash_error(
"The ideal %s is the unit ideal, please specify a different modulus.",
'(' + modulus + ')')
m = 0
info['modulus'] = m
# Hack to reduce polynomials and to handle non integral stuff
def redc(c):
return m.reduce(c * c.denominator()) / m.reduce(c.denominator())
def redp(f):
c = f.dict()
return f.parent()(dict((e, redc(c[e])) for e in c))
def safe_reduce(f):
if not m:
return latex(f)
try:
if f in sam.field():
return latex(redc(f))
else:
return latex(redp(f))
except ZeroDivisionError:
return '\\textrm{Unable to reduce} \\bmod\\mathfrak{m}'
info['reduce'] = safe_reduce
else:
info['reduce'] = latex
# check that explicit formula is not ridiculously big
if sam.explicit_formula():
info['explicit_formula_bytes'] = len(sam.explicit_formula())
if len(sam.explicit_formula()) < 100000:
info['explicit_formula'] = sam.explicit_formula()
return render_template("ModularForm_GSp4_Q_sample.html",
title=title,
bread=bread,
properties2=properties,
info=info)
def nf_string_to_label(F): # parse Q, Qsqrt2, Qsqrt-4, Qzeta5, etc
if F == 'Q':
return '1.1.1.1'
if F == 'Qi' or F == 'Q(i)':
return '2.0.4.1'
# Change unicode dash with minus sign
F = F.replace(u'\u2212', '-')
# remove non-ascii characters from F
F = F.decode('utf8').encode('ascii', 'ignore')
if len(F) == 0:
raise ValueError(
"Entry for the field was left blank. You need to enter a field label, field name, or a polynomial."
)
if F[0] == 'Q':
if '(' in F and ')' in F:
F = F.replace('(', '').replace(')', '')
if F[1:5] in ['sqrt', 'root']:
try:
d = ZZ(str(F[5:])).squarefree_part()
except (TypeError, ValueError):
d = 0
if d == 0:
raise ValueError(
"After {0}, the remainder must be a nonzero integer. Use {0}5 or {0}-11 for example."
.format(F[:5]))
if d == 1:
return '1.1.1.1'
if d % 4 in [2, 3]:
D = 4 * d
else:
D = d
absD = D.abs()
s = 0 if D < 0 else 2
return '2.%s.%s.1' % (s, str(absD))
if F[1:5] == 'zeta':
if '_' in F:
F = F.replace('_', '')
try:
d = ZZ(str(F[5:]))
except ValueError:
d = 0
if d < 1:
raise ValueError(
"After {0}, the remainder must be a positive integer. Use {0}5 for example."
.format(F[:5]))
if d % 4 == 2:
d /= 2 # Q(zeta_6)=Q(zeta_3), etc)
if d == 1:
return '1.1.1.1'
deg = euler_phi(d)
if deg > 23:
raise ValueError('%s is not in the database.' % F)
adisc = CyclotomicField(d).discriminant().abs() # uses formula!
return '%s.0.%s.1' % (deg, adisc)
raise ValueError(
'It is not a valid field name or label, or a defining polynomial.')
# check if a polynomial was entered
F = F.replace('X', 'x')
if 'x' in F:
F1 = F.replace('^', '**')
# print F
from lmfdb.number_fields.number_field import poly_to_field_label
F1 = poly_to_field_label(F1)
if F1:
return F1
raise ValueError('%s does not define a number field in the database.' %
F)
# Expand out factored labels, like 11.11.11e20.1
if not re.match(r'\d+\.\d+\.[0-9e_]+\.\d+', F):
raise ValueError("It must be of the form d.r.D.n, such as 2.2.5.1.")
parts = F.split(".")
def raise_power(ab):
if ab.count("e") == 0:
return ZZ(ab)
elif ab.count("e") == 1:
a, b = ab.split("e")
return ZZ(a)**ZZ(b)
else:
raise ValueError(
"Malformed absolute discriminant. It must be a sequence of strings AeB for A and B integers, joined by _s. For example, 2e7_3e5_11."
)
parts[2] = str(prod(raise_power(c) for c in parts[2].split("_")))
return ".".join(parts)
def nf_string_to_label(FF): # parse Q, Qsqrt2, Qsqrt-4, Qzeta5, etc
if FF in ['q', 'Q']:
return '1.1.1.1'
if FF.lower() in ['qi', 'q(i)']:
return '2.0.4.1'
# Change unicode dash with minus sign
FF = FF.replace(u'\u2212', '-')
# remove non-ascii characters from F
# we need to encode and decode for Python 3, as 'str' object has no attribute 'decode'
FF = FF.encode('utf8').decode('utf8').encode('ascii', 'ignore')
F = FF.lower() # keep original if needed
if len(F) == 0:
raise SearchParsingError(
"Entry for the field was left blank. You need to enter a field label, field name, or a polynomial."
)
if F[0] == 'q':
if '(' in F and ')' in F:
F = F.replace('(', '').replace(')', '')
if F[1:5] in ['sqrt', 'root']:
try:
d = ZZ(str(F[5:])).squarefree_part()
except (TypeError, ValueError):
d = 0
if d == 0:
raise SearchParsingError(
"After {0}, the remainder must be a nonzero integer. Use {0}5 or {0}-11 for example."
.format(FF[:5]))
if d == 1:
return '1.1.1.1'
if d % 4 in [2, 3]:
D = 4 * d
else:
D = d
absD = D.abs()
s = 0 if D < 0 else 2
return '2.%s.%s.1' % (s, str(absD))
if F[0:5] == 'qzeta':
if '_' in F:
F = F.replace('_', '')
match_obj = re.match(r'^qzeta(\d+)(\+|plus)?$', F)
if not match_obj:
raise SearchParsingError(
"After {0}, the remainder must be a positive integer or a positive integer followed by '+'. Use {0}5 or {0}19+, for example."
.format(F[:5]))
d = ZZ(str(match_obj.group(1)))
if d % 4 == 2:
d /= 2 # Q(zeta_6)=Q(zeta_3), etc)
if match_obj.group(2): # asking for the totally real field
from lmfdb.number_fields.web_number_field import rcyclolookup
if d in rcyclolookup:
return rcyclolookup[d]
else:
raise SearchParsingError('%s is not in the database.' % F)
# Now not the totally real subfield
from lmfdb.number_fields.web_number_field import cyclolookup
if d in cyclolookup:
return cyclolookup[d]
else:
raise SearchParsingError('%s is not in the database.' % F)
raise SearchParsingError(
'It is not a valid field name or label, or a defining polynomial.')
# check if a polynomial was entered
F = F.replace('X', 'x')
if 'x' in F:
F1 = F.replace('^', '**')
# print F
from lmfdb.number_fields.number_field import poly_to_field_label
F1 = poly_to_field_label(F1)
if F1:
return F1
raise SearchParsingError(
'%s does not define a number field in the database.' % F)
# Expand out factored labels, like 11.11.11e20.1
if not re.match(r'\d+\.\d+\.[0-9e_]+\.\d+', F):
raise SearchParsingError(
"A number field label must be of the form d.r.D.n, such as 2.2.5.1."
)
parts = F.split(".")
def raise_power(ab):
if ab.count("e") == 0:
return ZZ(ab)
elif ab.count("e") == 1:
a, b = ab.split("e")
return ZZ(a)**ZZ(b)
else:
raise SearchParsingError(
"Malformed absolute discriminant. It must be a sequence of strings AeB for A and B integers, joined by _s. For example, 2e7_3e5_11."
)
parts[2] = str(prod(raise_power(c) for c in parts[2].split("_")))
return ".".join(parts)
def nf_string_to_label(F): # parse Q, Qsqrt2, Qsqrt-4, Qzeta5, etc
if F == 'Q':
return '1.1.1.1'
if F == 'Qi' or F == 'Q(i)':
return '2.0.4.1'
# Change unicode dash with minus sign
F = F.replace(u'\u2212', '-')
# remove non-ascii characters from F
F = F.decode('utf8').encode('ascii', 'ignore')
if len(F) == 0:
raise ValueError("Entry for the field was left blank. You need to enter a field label, field name, or a polynomial.")
if F[0] == 'Q':
if '(' in F and ')' in F:
F=F.replace('(','').replace(')','')
if F[1:5] in ['sqrt', 'root']:
try:
d = ZZ(str(F[5:])).squarefree_part()
except (TypeError, ValueError):
d = 0
if d == 0:
raise ValueError("After {0}, the remainder must be a nonzero integer. Use {0}5 or {0}-11 for example.".format(F[:5]))
if d == 1:
return '1.1.1.1'
if d % 4 in [2, 3]:
D = 4 * d
else:
D = d
absD = D.abs()
s = 0 if D < 0 else 2
return '2.%s.%s.1' % (s, str(absD))
if F[1:5] == 'zeta':
if '_' in F:
F = F.replace('_','')
try:
d = ZZ(str(F[5:]))
except ValueError:
d = 0
if d < 1:
raise ValueError("After {0}, the remainder must be a positive integer. Use {0}5 for example.".format(F[:5]))
if d % 4 == 2:
d /= 2 # Q(zeta_6)=Q(zeta_3), etc)
if d == 1:
return '1.1.1.1'
deg = euler_phi(d)
if deg > 23:
raise ValueError('%s is not in the database.' % F)
adisc = CyclotomicField(d).discriminant().abs() # uses formula!
return '%s.0.%s.1' % (deg, adisc)
raise ValueError('It is not a valid field name or label, or a defining polynomial.')
# check if a polynomial was entered
F = F.replace('X', 'x')
if 'x' in F:
F1 = F.replace('^', '**')
# print F
from lmfdb.number_fields.number_field import poly_to_field_label
F1 = poly_to_field_label(F1)
if F1:
return F1
raise ValueError('%s does not define a number field in the database.'%F)
# Expand out factored labels, like 11.11.11e20.1
if not re.match(r'\d+\.\d+\.[0-9e_]+\.\d+',F):
raise ValueError("It must be of the form d.r.D.n, such as 2.2.5.1.")
parts = F.split(".")
def raise_power(ab):
if ab.count("e") == 0:
return ZZ(ab)
elif ab.count("e") == 1:
a,b = ab.split("e")
return ZZ(a)**ZZ(b)
else:
raise ValueError("Malformed absolute discriminant. It must be a sequence of strings AeB for A and B integers, joined by _s. For example, 2e7_3e5_11.")
parts[2] = str(prod(raise_power(c) for c in parts[2].split("_")))
return ".".join(parts)
def render_sample_page(family, sam, args, bread):
info = { 'args': to_dict(args), 'sam': sam, 'latex': latex, 'type':sam.type(), 'name':sam.name(), 'full_name': sam.full_name(), 'weight':sam.weight(), 'fdeg':sam.degree_of_field(), 'is_eigenform':sam.is_eigenform(), 'field_poly': sam.field_poly()}
if sam.is_integral() != None:
info['is_integral'] = sam.is_integral()
if 'Sp4Z' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=0)
if 'Sp4Z_2' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=2)
info['space'] = '$'+family.latex_name.replace('k', '{' + str(sam.weight()) + '}')+'$'
if 'space_url' in info:
bread.append((info['space'], info['space_url']))
info['space_href'] = '<a href="%s">%s</d>'%(info['space_url'],info['space']) if 'space_url' in info else info['space']
if info['field_poly'].disc() < 10**10:
label = poly_to_field_label(info['field_poly'])
if label:
info['field_label'] = label
info['field_url'] = url_for('number_fields.by_label', label=label)
info['field_href'] = '<a href="%s">%s</a>'%(info['field_url'], field_pretty(label))
bread.append((info['name'], ''))
title='Siegel modular forms sample ' + info['full_name']
properties = [('Space', info['space_href']),
('Name', info['name']),
('Type', '<br>'.join(info['type'].split(','))),
('Weight', str(info['weight'])),
('Hecke eigenform', str(info['is_eigenform'])),
('Field degree', str(info['fdeg']))]
try:
evs_to_show = parse_ints_to_list_flash(args.get('ev_index'), 'list of $l$')
fcs_to_show = parse_ints_to_list_flash(args.get('fc_det'), 'list of $\\det(F)$')
except ValueError:
evs_to_show = []
fcs_to_show = []
info['evs_to_show'] = sorted([n for n in (evs_to_show if len(evs_to_show) else sam.available_eigenvalues()[:10])])
info['fcs_to_show'] = sorted([n for n in (fcs_to_show if len(fcs_to_show) else sam.available_Fourier_coefficients()[1:6])])
info['evs_avail'] = [n for n in sam.available_eigenvalues()]
info['fcs_avail'] = [n for n in sam.available_Fourier_coefficients()]
# Do not attempt to constuct a modulus ideal unless the field has a reasonably small discriminant
# otherwise sage may not even be able to factor the discriminant
info['field'] = sam.field()
if info['field_poly'].disc() < 10**80:
null_ideal = sam.field().ring_of_integers().ideal(0)
info['modulus'] = null_ideal
modulus = args.get('modulus','').strip()
m = 0
if modulus:
try:
O = sam.field().ring_of_integers()
m = O.ideal([O(str(b)) for b in modulus.split(',')])
except Exception:
info['error'] = True
flash_error("Unable to construct modulus ideal from specified generators %s.", modulus)
if m == 1:
info['error'] = True
flash_error("The ideal %s is the unit ideal, please specify a different modulus.", '('+modulus+')')
m = 0
info['modulus'] = m
# Hack to reduce polynomials and to handle non integral stuff
def redc(c):
return m.reduce(c*c.denominator())/m.reduce(c.denominator())
def redp(f):
c = f.dict()
return f.parent()(dict((e,redc(c[e])) for e in c))
def safe_reduce(f):
if not m:
return latex(f)
try:
if f in sam.field():
return latex(redc(f))
else:
return latex(redp(f))
except ZeroDivisionError:
return '\\textrm{Unable to reduce} \\bmod\\mathfrak{m}'
info['reduce'] = safe_reduce
else:
info['reduce'] = latex
# check that explicit formula is not ridiculously big
if sam.explicit_formula():
info['explicit_formula_bytes'] = len(sam.explicit_formula())
if len(sam.explicit_formula()) < 100000:
info['explicit_formula'] = sam.explicit_formula()
return render_template("ModularForm_GSp4_Q_sample.html", title=title, bread=bread, properties2=properties, info=info)
def nf_string_to_label(F): # parse Q, Qsqrt2, Qsqrt-4, Qzeta5, etc
if F == "Q":
return "1.1.1.1"
if F == "Qi":
return "2.0.4.1"
# Change unicode dash with minus sign
F = F.replace(u"\u2212", "-")
# remove non-ascii characters from F
F = F.decode("utf8").encode("ascii", "ignore")
fail_string = str(F + " is not a valid field label or name or polynomial, or is not ")
if len(F) == 0:
raise ValueError(
"Entry for the field was left blank. You need to enter a field label, field name, or a polynomial."
)
if F[0] == "Q":
if F[1:5] in ["sqrt", "root"]:
try:
d = ZZ(str(F[5:])).squarefree_part()
except ValueError:
d = 0
if d == 0:
raise ValueError(
"After {0}, the remainder must be a nonzero integer. Use {0}5 or {0}-11 for example.".format(F[:5])
)
if d % 4 in [2, 3]:
D = 4 * d
else:
D = d
absD = D.abs()
s = 0 if D < 0 else 2
return "2.%s.%s.1" % (s, str(absD))
if F[1:5] == "zeta":
try:
d = ZZ(str(F[5:]))
except ValueError:
d = 0
if d < 1:
raise ValueError(
"After {0}, the remainder must be a positive integer. Use {0}5 for example.".format(F[:5])
)
if d % 4 == 2:
d /= 2 # Q(zeta_6)=Q(zeta_3), etc)
if d == 1:
return "1.1.1.1"
deg = euler_phi(d)
if deg > 23:
raise ValueError("%s is not in the database." % F)
adisc = CyclotomicField(d).discriminant().abs() # uses formula!
return "%s.0.%s.1" % (deg, adisc)
return fail_string
# check if a polynomial was entered
F = F.replace("X", "x")
if "x" in F:
F1 = F.replace("^", "**")
# print F
from lmfdb.number_fields.number_field import poly_to_field_label
F1 = poly_to_field_label(F1)
if F1:
return F1
raise ValueError("%s is not in the database." % F)
# Expand out factored labels, like 11.11.11e20.1
parts = F.split(".")
if len(parts) != 4:
raise ValueError("It must be of the form <deg>.<real_emb>.<absdisc>.<number>, such as 2.2.5.1.")
def raise_power(ab):
if ab.count("e") == 0:
return ZZ(ab)
elif ab.count("e") == 1:
a, b = ab.split("e")
return ZZ(a) ** ZZ(b)
else:
raise ValueError(
"Malformed absolute discriminant. It must be a sequence of strings AeB for A and B integers, joined by _s. For example, 2e7_3e5_11."
)
parts[2] = str(prod(raise_power(c) for c in parts[2].split("_")))
return ".".join(parts)