def get_number_field_integral_basis(c_string):
r"""Get the integral basis for the field specified by the string.
INPUT:
c_string -- string, a string of comma-separated coefficients with no spaces: the coefficients of the normalized (using gp.polredabs) defining polynomial
OUTPUT:
fld_bool -- bool, True if the number field has a page in the LMFDB, False otherwise
K_new -- number field, the number field with defining polynomial that is the normalized version (given by gp.polredabs) of the one with coefficients specified by c_string
a -- number field element, generator for K_new
the integral basis for K_new recorded on its LMFDB page
"""
C = getDBconnection()
c_hash = field_coeffs_string_to_hash(c_string)
field = C.numberfields.fields.find_one({'coeffhash': c_hash})
fld_bool = True
try:
field['degree']
except TypeError:
fld_bool = False
if fld_bool:
field_str = field['coeffs']
int_basis_str = field['zk']
poly = coeffs_to_poly(field_str)
K_new = NumberField(poly, names=('a', ))
(a, ) = K_new._first_ngens(1)
return fld_bool, K_new, a, [
K_new(eval(preparse(el))) for el in int_basis_str
]
else:
# could add polynomial to list of number fields missing from LMFDB here
return fld_bool, None, None, None
def get_number_field_integral_basis(c_string):
r"""Get the integral basis for the field specified by the string.
INPUT:
c_string -- string, a string of comma-separated coefficients with no spaces: the coefficients of the normalized (using gp.polredabs) defining polynomial
OUTPUT:
fld_bool -- bool, True if the number field has a page in the LMFDB, False otherwise
K_new -- number field, the number field with defining polynomial that is the normalized version (given by gp.polredabs) of the one with coefficients specified by c_string
a -- number field element, generator for K_new
the integral basis for K_new recorded on its LMFDB page
"""
C = getDBconnection()
c_hash = field_coeffs_string_to_hash(c_string)
field = C.numberfields.fields.find_one({'coeffhash':c_hash})
fld_bool = True
try:
field['degree']
except TypeError:
fld_bool = False
if fld_bool:
field_str = field['coeffs']
int_basis_str = field['zk']
poly = coeffs_to_poly(field_str)
K_new = NumberField(poly, names=('a',))
(a,) = K_new._first_ngens(1)
return fld_bool, K_new, a, [K_new(eval(preparse(el))) for el in int_basis_str]
else:
# could add polynomial to list of number fields missing from LMFDB here
return fld_bool, None, None, None
def get_hmfs_hecke_field_and_eigenvals(label):
"""Get the Hecke field and eigenvalues for the Hilbert modular form with given label.
INPUT:
label -- string, the label of the Hilbert modular form
OUTPUT:
K_old -- number field, the field containing the Hecke eigenvalues
e -- number field element, a generator for K_old over QQ
eigenvals -- list, a list of the Hecke eigenvalues
"""
C = getDBconnection()
# Should I use find_one, or something else?
R = PolynomialRing(QQ, names=('x'))
form = C.hmfs.forms.find_one({'label': label})
poly = R(str(form['hecke_polynomial']))
K_old = NumberField(poly, names=('e', ))
(e, ) = K_old._first_ngens(1)
eigenvals_str = form['hecke_eigenvalues']
eigenvals = [K_old(eval(preparse(el))) for el in eigenvals_str]
return K_old, e, eigenvals
def get_hmfs_hecke_field_and_eigenvals(label):
"""Get the Hecke field and eigenvalues for the Hilbert modular form with given label.
INPUT:
label -- string, the label of the Hilbert modular form
OUTPUT:
K_old -- number field, the field containing the Hecke eigenvalues
e -- number field element, a generator for K_old over QQ
eigenvals -- list, a list of the Hecke eigenvalues
"""
C = getDBconnection()
# Should I use find_one, or something else?
R = PolynomialRing(QQ,names=('x'))
form = C.hmfs.forms.find_one({'label':label})
poly = R(str(form['hecke_polynomial']))
K_old = NumberField(poly, names=('e',))
(e,) = K_old._first_ngens(1)
eigenvals_str = form['hecke_eigenvalues']
eigenvals = [K_old(eval(preparse(el))) for el in eigenvals_str]
return K_old, e, eigenvals
