def see_frobs(frob_data):
ans = []
seeram = False
plist = [p for p in primes(2, 60)]
for i in range(len(plist)):
p = plist[i]
dec = frob_data[i][1]
if dec[0] == 0:
ans.append([p, 'R'])
seeram = True
else:
s = '$'
firstone = True
for j in dec:
if not firstone:
s += r'{,}\,'
if j[0] < 15:
s += r'{\href{%s}{%d} }' % (url_for(
'local_fields.by_label', label="%d.%d.0.1" %
(p, j[0])), j[0])
else:
s += str(j[0])
if j[1] > 1:
s += '^{' + str(j[1]) + '}'
firstone = False
s += '$'
ans.append([p, s])
return ans, seeram
def frobs(nf):
frob_at_p = residue_field_degrees_function(nf)
D = nf.disc()
ans = []
seeram = False
for p in primes(2, 60):
if not ZZ(p).divides(D):
# [3] , [2,1]
dec = frob_at_p(p)
vals = list(set(dec))
vals = sorted(vals, reverse=True)
dec = [[x, dec.count(x)] for x in vals]
#dec2 = ["$" + str(x[0]) + ('^{' + str(x[1]) + '}$' if x[1] > 1 else '$') for x in dec]
s = '$'
firstone = 1
for j in dec:
if firstone == 0:
s += '{,}\,'
s += str(j[0])
if j[1] > 1:
s += '^{' + str(j[1]) + '}'
firstone = 0
s += '$'
ans.append([p, s])
else:
ans.append([p, 'R'])
seeram = True
return ans, seeram
def frobs(nf):
frob_at_p = residue_field_degrees_function(nf)
D = nf.disc()
ans = []
seeram = False
for p in primes(2, 60):
if not ZZ(p).divides(D):
# [3] , [2,1]
dec = frob_at_p(p)
vals = list(set(dec))
vals = sorted(vals, reverse=True)
dec = [[x, dec.count(x)] for x in vals]
#dec2 = ["$" + str(x[0]) + ('^{' + str(x[1]) + '}$' if x[1] > 1 else '$') for x in dec]
s = '$'
firstone = 1
for j in dec:
if firstone == 0:
s += '{,}\,'
if j[0] < 15:
s += r'{\href{%s}{%d} }' % (url_for(
'local_fields.by_label', label="%d.%d.0.1" %
(p, j[0])), j[0])
else:
s += str(j[0])
if j[1] > 1:
s += '^{' + str(j[1]) + '}'
firstone = 0
s += '$'
ans.append([p, s])
else:
ans.append([p, 'R'])
seeram = True
return ans, seeram
def frobs(K):
frob_at_p = residue_field_degrees_function(K)
D = K.disc()
ans = []
seeram = False
for p in primes(2, 60):
if not ZZ(p).divides(D):
# [3] , [2,1]
dec = frob_at_p(p)
vals = list(set(dec))
vals = sorted(vals, reverse=True)
dec = [[x, dec.count(x)] for x in vals]
dec2 = ["$" + str(x[0]) + ('^{' + str(x[1]) + '}$' if x[1] > 1 else '$') for x in dec]
s = '$'
firstone = 1
for j in dec:
if firstone == 0:
s += '{,}\,'
s += str(j[0])
if j[1] > 1:
s += '^{' + str(j[1]) + '}'
firstone = 0
s += '$'
ans.append([p, s])
else:
ans.append([p, 'R'])
seeram = True
return ans, seeram
def frobs(K):
frob_at_p = residue_field_degrees_function(K)
D = K.disc()
ans = []
seeram = False
for p in primes(2, 60):
if not ZZ(p).divides(D):
# [3] , [2,1]
dec = frob_at_p(p)
vals = list(set(dec))
vals = sorted(vals, reverse=True)
dec = [[x, dec.count(x)] for x in vals]
dec2 = ["$" + str(x[0]) + ("^{" + str(x[1]) + "}$" if x[1] > 1 else "$") for x in dec]
s = "$"
firstone = 1
for j in dec:
if firstone == 0:
s += "{,}\,"
s += str(j[0])
if j[1] > 1:
s += "^{" + str(j[1]) + "}"
firstone = 0
s += "$"
ans.append([p, s])
else:
ans.append([p, "R"])
seeram = True
return ans, seeram
def test1(B=10**4):
from sage.all import polygen, primes, QQ, EllipticCurve
import psage.libs.smalljac.wrapper
x = polygen(QQ, 'x')
J = psage.libs.smalljac.wrapper.SmallJac(x**3 + 17 * x + 3)
E = EllipticCurve([17, 3])
N = E.conductor()
for p in primes(B):
if N % p:
assert E.ap(p) == J.ap(p)
def test2(B=500):
from sage.all import polygen, primes, QQ, GF, HyperellipticCurve
import psage.libs.smalljac.wrapper
x = polygen(QQ, 'x')
J = psage.libs.smalljac.wrapper.SmallJac(x**5 + 17 * x + 3)
N = 97 * 3749861
for p in primes(45, B): # 45 comes from "(2g+1)(2N-1) = 45"
if N % p:
x = polygen(GF(p), 'x')
C = HyperellipticCurve(x**5 + 17 * x + 3)
assert C.frobenius_polynomial() == J.frob(p).charpoly()
def test1(B=10 ** 4):
from sage.all import polygen, primes, QQ, EllipticCurve
import psage.libs.smalljac.wrapper
x = polygen(QQ, "x")
J = psage.libs.smalljac.wrapper.SmallJac(x ** 3 + 17 * x + 3)
E = EllipticCurve([17, 3])
N = E.conductor()
for p in primes(B):
if N % p:
assert E.ap(p) == J.ap(p)
def test2(B=500):
from sage.all import polygen, primes, QQ, GF, HyperellipticCurve
import psage.libs.smalljac.wrapper
x = polygen(QQ, "x")
J = psage.libs.smalljac.wrapper.SmallJac(x ** 5 + 17 * x + 3)
N = 97 * 3749861
for p in primes(45, B): # 45 comes from "(2g+1)(2N-1) = 45"
if N % p:
x = polygen(GF(p), "x")
C = HyperellipticCurve(x ** 5 + 17 * x + 3)
assert C.frobenius_polynomial() == J.frob(p).charpoly()
def read_line(line, debug=0):
r""" Parses one line from input file. Returns and a dict containing
fields with keys as above.
Sample line: 11 a 1 0,-1,1,-10,-20 7 1,0 0,1,0 0,0 0,1
Fields: label (3 fields)
a-invariants
p0
For each bad prime: 'a' if additive
lambda,mu if multiplicative (or 'o?' if unknown)
For each good prime: lambda,mu if ordinary (or 'o?' if unknown)
lambda+,lambda-,mu if supersingular (or 's?' if unknown)
"""
data = {}
if debug: print("Parsing input line {}".format(line[:-1]))
fields = line.split()
label = fields[0]+fields[1]+fields[2]
data['label'] = label
N = ZZ(fields[0])
badp = N.support()
nbadp = len(badp)
p0 = int(fields[4])
data['iwp0'] = p0
if debug: print("p0={}".format(p0))
iwdata = {}
# read data for bad primes
for p,pdat in zip(badp,fields[5:5+nbadp]):
p = str(p)
if debug>1: print("p={}, pdat={}".format(p,pdat))
if pdat in ['o?','a']:
iwdata[p]=pdat
else:
iwdata[p]=[int(x) for x in pdat.split(",")]
# read data for all primes
for p,pdat in zip(primes(1000),fields[5+nbadp:]):
p = str(p)
if debug>1: print("p={}, pdat={}".format(p,pdat))
if pdat in ['s?','o?','a']:
iwdata[p]=pdat
else:
iwdata[p]=[int(x) for x in pdat.split(",")]
data['iwdata'] = iwdata
if debug: print("label {}, data {}".format(label,data))
return label, data
def read_line(line, debug=0):
r""" Parses one line from input file. Returns and a dict containing
fields with keys as above.
Sample line: 11 a 1 0,-1,1,-10,-20 7 1,0 0,1,0 0,0 0,1
Fields: label (3 fields)
a-invariants
p0
For each bad prime: 'a' if additive
lambda,mu if multiplicative (or 'o?' if unknown)
For each good prime: lambda,mu if ordinary (or 'o?' if unknown)
lambda+,lambda-,mu if supersingular (or 's?' if unknown)
"""
data = {}
if debug: print("Parsing input line {}".format(line[:-1]))
fields = line.split()
label = fields[0] + fields[1] + fields[2]
data['label'] = label
N = ZZ(fields[0])
badp = N.support()
nbadp = len(badp)
p0 = int(fields[4])
data['iwp0'] = p0
if debug: print("p0={}".format(p0))
iwdata = {}
# read data for bad primes
for p, pdat in zip(badp, fields[5:5 + nbadp]):
p = str(p)
if debug > 1: print("p={}, pdat={}".format(p, pdat))
if pdat in ['o?', 'a']:
iwdata[p] = pdat
else:
iwdata[p] = [int(x) for x in pdat.split(",")]
# read data for all primes
for p, pdat in zip(primes(1000), fields[5 + nbadp:]):
p = str(p)
if debug > 1: print("p={}, pdat={}".format(p, pdat))
if pdat in ['s?', 'o?', 'a']:
iwdata[p] = pdat
else:
iwdata[p] = [int(x) for x in pdat.split(",")]
data['iwdata'] = iwdata
if debug: print("label {}, data {}".format(label, data))
return label, data
def period_mapping(M):
N=M.level()
S=M.cuspidal_subspace()
q=exists(primes(N+1,N**3+3), lambda x: x%N==1)[1]
#q is the smallest prime equal to 1 mod N
Tq=M.hecke_matrix(q)
dq=M.diamond_bracket_operator(q).matrix()
to_cusp_space=(Tq-dq*q-1) #maybe use smaller primes in the future*(Tq-dq-q)
#to_cusp_space is a map whose image (over QQ) is the cuspidal subspace and is invertible when
#restricted to the cuspidal subspace so this means that we can compute the intergral period
#mapping (projection onto the cusps) by composing to_cusp_space by its inverse on the cuspidal
#subspace
codomain_restricted = to_cusp_space.restrict_codomain(S)
restricted = codomain_restricted.restrict_domain(S)
return codomain_restricted*restricted**(-1)
def period_mapping(M):
N=M.level()
S=M.cuspidal_subspace()
q=exists(primes(N+1,N**3+3), lambda x: x%N==1)[1]
#q is the smallest prime equal to 1 mod N
Tq=M.hecke_matrix(q)
dq=M.diamond_bracket_operator(q).matrix()
to_cusp_space=(Tq-dq*q-1) #maybe use smaller primes in the future*(Tq-dq-q)
#to_cusp_space is a map whose image (over QQ) is the cuspidal subspace and is invertible when
#restricted to the cuspidal subspace so this means that we can compute the intergral period
#mapping (projection onto the cusps) by composing to_cusp_space by its inverse on the cuspidal
#subspace
codomain_restricted = to_cusp_space.restrict_codomain(S)
restricted = codomain_restricted.restrict_domain(S)
return codomain_restricted*restricted**(-1)
def compute_local_roots_SMF2_scalar_valued(K, ev, k, embedding):
''' computes the dirichlet series for a Lfunction_SMF2_scalar_valued
'''
L = ev.keys()
m = ZZ(max(L)).isqrt() + 1
ev2 = {}
for p in primes(m):
try:
ev2[p] = (ev[p], ev[p * p])
except:
break
logger.debug(str(ev2))
ret = []
for p in ev2:
R = PolynomialRing(K, 'x')
x = R.gens()[0]
f = (1 - ev2[p][0] * x +
(ev2[p][0]**2 - ev2[p][1] - p**(2 * k - 4)) * x**2 -
ev2[p][0] * p**(2 * k - 3) * x**3 + p**(4 * k - 6) * x**4)
Rnum = PolynomialRing(CF, 'y')
x = Rnum.gens()[0]
fnum = Rnum(0)
if K != QQ:
for i in range(int(f.degree()) + 1):
fnum = fnum + f[i].complex_embeddings(NN)[embedding] * (
x / p**(k - 1.5))**i
else:
for i in range(int(f.degree()) + 1):
fnum = fnum + f[i] * (x / CF(p**(k - 1.5)))**i
r = fnum.roots(CF)
r = [1 / a[0] for a in r]
# a1 = r[1][0]/r[0][0]
# a2 = r[2][0]/r[0][0]
# a0 = 1/r[3][0]
ret.append((p, r))
return ret
def compute_local_roots_SMF2_scalar_valued(K, ev, k, embedding):
''' computes the dirichlet series for a Lfunction_SMF2_scalar_valued
'''
L = ev.keys()
m = ZZ(max(L)).isqrt() + 1
ev2 = {}
for p in primes(m):
try:
ev2[p] = (ev[p], ev[p * p])
except:
break
logger.debug(str(ev2))
ret = []
for p in ev2:
R = PolynomialRing(K, 'x')
x = R.gens()[0]
f = (1 - ev2[p][0] * x + (ev2[p][0] ** 2 - ev2[p][1] - p ** (
2 * k - 4)) * x ** 2 - ev2[p][0] * p ** (2 * k - 3) * x ** 3 + p ** (4 * k - 6) * x ** 4)
Rnum = PolynomialRing(CF, 'y')
x = Rnum.gens()[0]
fnum = Rnum(0)
if K != QQ:
for i in range(int(f.degree()) + 1):
fnum = fnum + f[i].complex_embeddings(NN)[embedding] * (x / p ** (k - 1.5)) ** i
else:
for i in range(int(f.degree()) + 1):
fnum = fnum + f[i] * (x / CF(p ** (k - 1.5))) ** i
r = fnum.roots(CF)
r = [1 / a[0] for a in r]
# a1 = r[1][0]/r[0][0]
# a2 = r[2][0]/r[0][0]
# a0 = 1/r[3][0]
ret.append((p, r))
return ret
def get_form_data_from_file(nf, data_dir=NF_DIR, max_dim=50, verbose=False):
"""Fill in data for an incomplete WebNewform object by reading from a
data file. If a suitable file does not exist, the original
WebNewform object is returned unchanged, with a message output.
The data files only contain the extra data for dimensions less
than some bound. If the newform's dimension is greater then we do
nothing.
"""
if not nf.field_poly is None:
return nf
fname = label = nf.label
datafile = "/".join([data_dir, fname])
try:
data = read_dtp(datafile, verbose=False)
except FileNotFoundError:
fname = fname[:fname.rindex(".")]
datafile = "/".join([data_dir, fname])
try:
data = read_dtp(datafile, verbose=False)
except FileNotFoundError:
print("No file {} or {} found: no data available".format(datafile,label))
return nf
if verbose:
print("Successfully read data for {} from file {}".format(label, fname))
# Now we have data. It is a dict with a single key (N,k,o) where
# o is the character orbit number and value so we just extract the
# single value for this key, which is another dict:
data = list(data.values())[0]
# This dict has keys ['dims', 'traces', 'ALs', 'polys',
# 'eigdata'], and each value is a list, one per newform, so we
# need to extract just one item from the relevant lists, according
# to the newform we want. This is determined by the 4th (last)
# part of the label we have, which we need to convert to an
# integer (from 0).
# NB eigdata is a list of dicts (see later comment for details)
# but *only* for components of dimension>1
nf_index = class_to_int(label.split(".")[3])
dims = data['dims']
nf_eigdata_index = nf_index - dims.count(1)
if verbose: # debug
print("Newform label = {}".format(label))
print("nf_index = {}".format(nf_index))
print("len(polys) = {}".format(len(data['polys'])))
print("len(dims) = {}".format(len(dims)))
print("#dims>1 = {}".format(len(dims)-dims.count(1)))
print("len(eigdata) = {}".format(len(data['eigdata'])))
print("nf_eigdata_index = {}".format(nf_eigdata_index))
dim = data['dims'][nf_index]
assert dim == nf.dim
nf.field_poly = data['polys'][nf_index]
eigdata = data['eigdata'][nf_eigdata_index]
# eigdata is a dict with keys ['poly', 'basis', 'n', 'm', 'ans', 'char']
chi_gens, chi_vals = eigdata['char']
nf.hecke_ring_character_values = list(zip(chi_gens, chi_vals))
# we will not need to access the char_order since this space has
# already been selected to have an appropriate character.
if verbose:
print("hecke_ring_character_values = {}".format(nf.hecke_ring_character_values))
Qx = PolynomialRing(QQ,'x')
nf.hecke_field = K = NumberField(Qx(nf.field_poly), 'a_')
nf.betas = [K(b) for b in eigdata['basis']]
nf.an = eigdata['ans']
nf.ap = [nf.an[p-1] for p in primes(len(nf.an))]
if verbose: # debug
print("We have {} a_n and {} a_p for newforms {}".format(len(nf.an), len(nf.ap), label))
return nf
def make_euler_factors_db(E):
r"""
Returns a list of the Euler factors for all primes up to 100,
given a database elliptic curve E (which has this many ap stored)
"""
return [make_one_euler_factor_db(E, p) for p in primes(100)]
def make_euler_factors(E, maxp=100):
r"""
Returns a list of the Euler factors for all primes up to max_p,
given a Sage elliptic curve E.
"""
return [make_one_euler_factor(E, p) for p in primes(maxp)]
def prime_powers(B=100, emax=5):
for p in primes(B+1):
for P in F.primes_above(p):
for e in range(1,emax+1):
yield p, P, e
def prime_powers(B=100, emax=5):
for p in primes(B + 1):
for P in F.primes_above(p):
for e in range(1, emax + 1):
yield p, P, e
