def main4():
print(crack_when_pq_close(23360947609))
p = next_prime(2**128)
print(p)
q = next_prime(p)
print(q)
print(crack_when_pq_close(p * q))
def generate_pseudoprime(bases, min_bitsize=0):
"""
Generates a pseudoprime which passes the Miller-Rabin primality test for the provided bases.
:param bases: the bases
:param min_bitsize: the minimum bitsize of the generated pseudoprime (default: 0)
:return: a tuple containing the pseudoprime, as well as its 3 prime factors
"""
bases.sort()
k2 = next_prime(bases[-1])
k3 = next_prime(k2)
while True:
residues = [inverse_mod(-k2, k3), inverse_mod(-k3, k2)]
moduli = [k3, k2]
s = _generate_s(bases, k2, k3)
residue, modulus = _backtrack(s, bases, residues, moduli, 0)
if residue and modulus:
i = (2 ** (min_bitsize // 3)) // modulus
while True:
p1 = residue + i * modulus
p2 = k2 * (p1 - 1) + 1
p3 = k3 * (p1 - 1) + 1
if is_prime(p1) and is_prime(p2) and is_prime(p3):
return p1 * p2 * p3, p1, p2, p3
i += 1
else:
k3 = next_prime(k3)
def main7():
set_random_seed(0)
p = next_prime(randrange(2**96))
q = next_prime(randrange(2**97))
n = p * q
print(p, q, n)
print(qsieve(n))
def make_hash(p, N1, N2, np=20):
plist = [next_prime(400000)]
while len(plist) < np:
plist.append(next_prime(plist[-1]))
hashtab = {}
nc = 0
for E in cremona_optimal_curves(srange(N1, N2 + 1)):
nc += 1
h = hash1(p, E, plist)
lab = E.label()
if nc % 1000 == 0:
print(lab)
#print("{} has hash = {}".format(lab,h))
if h in hashtab:
hashtab[h].append(lab)
#print("new set {}".format(hashtab[h]))
else:
hashtab[h] = [lab]
ns = 0
for s in hashtab.values():
if len(s) > 1:
ns += 1
#print s
print("{} sets of conjugate curves".format(ns))
return hashtab
def make_hash_many(plist, N1, N2, nq=20):
qlist = [next_prime(400000)]
while len(qlist) < nq:
qlist.append(next_prime(qlist[-1]))
hashtabs = dict([(p, dict()) for p in plist])
nc = 0
for E in cremona_optimal_curves(srange(N1, N2 + 1)):
nc += 1
lab = E.label()
if nc % 1000 == 0:
print(lab)
h = hash1many(plist, E, qlist)
for p in plist:
hp = h[p]
if hp in hashtabs[p]:
hashtabs[p][hp].append(lab)
print("p={}, new set {}".format(p, hashtabs[p][hp]))
else:
hashtabs[p][hp] = [lab]
ns = dict([(p, len([v for v in hashtabs[p].values() if len(v) > 1]))
for p in plist])
for p in plist:
print("p={}: {} nontrivial sets of congruent curves".format(p, ns[p]))
return hashtabs
def make_base_field(self):
if self.args.base_field != None:
F = GF(self.args.base_field, 'F')
self.field_traits.check(F)
return F
p = next_prime(2**self.args.num_bits)
while not self.field_traits.check(GF(p, 'F'), nothrow=True):
p = next_prime(p)
return GF(p, 'F')
def rsa(bits):
# only prove correctness up to 1024bits
proof = (bits <= 1024)
p = next_prime(ZZ.random_element(2**(bits // 2 + 1)),
proof=proof)
q = next_prime(ZZ.random_element(2**(bits // 2 + 1)),
proof=proof)
n = p * q
phi_n = (p - 1) * (q - 1)
while True:
e = ZZ.random_element(1, phi_n)
if gcd(e, phi_n) == 1:
break
d = lift(Mod(e, phi_n)**(-1))
return e, d, n
def make_hash_SW(plist, N1=0, N2=999, nq=30, tate=False):
"""Here N1 and N2 are in [0..999] and refer to the batch number, where
batch n has curves of conductor between n*10^5 and (n+1)*10^5
plist is a list of primes so we can make several hash tables in parallel
If tate=True then ignore curves unless ord_p(j(E))=-1
"""
# Compute the first nq primes greater than the largest conductor:
qlist = [next_prime((N2 + 1) * 10**5)]
while len(qlist) < nq:
qlist.append(next_prime(qlist[-1]))
# Initialize hash tables for each p:
hashtabs = dict([(p, dict()) for p in plist])
nc = 0
for N in range(N1, N2 + 1):
print("Batch {}".format(N))
for dat in SteinWatkinsAllData(N):
nc += 1
NE = dat.conductor
E = EllipticCurve(dat.curves[0][0])
assert E.conductor() == NE
if tate and all(E.j_invariant().valuation(p) >= 0 for p in plist):
continue
lab = SW_label(E)
if nc % 10000 == 0:
print(lab)
# if nc>10000:
# break
h = hash1many(plist, E, qlist)
for p in plist:
if tate and E.j_invariant().valuation(p) >= 0:
continue
hp = h[p]
if hp in hashtabs[p]:
hashtabs[p][hp].append(lab)
print("p={}, new set {}".format(p, hashtabs[p][hp]))
else:
hashtabs[p][hp] = [lab]
ns = dict([(p, len([v for v in hashtabs[p].values() if len(v) > 1]))
for p in plist])
for p in plist:
print("p={}: {} nontrivial sets of congruent curves".format(p, ns[p]))
return hashtabs
def __init__(self,
signature=0,
weight=2,
level_limit=34,
rank_limit=4,
primes=None,
simple_color=None,
nonsimple_color=None,
reduction=True,
bound=0):
"""
Initialize a SimpleModulesGraph containing finite quadratic modules of signature ``signature``.
They are checked for being ``weight``-simple if their minimal number of generators
is at most `rank_limit`.
INPUT:
- ``signature``: the signature of the modules
- ``weight``: check for cusp forms of weight ``weight``
- ``level_limit``: only check for anisotropic modules with level smaller than ``level_limit``
- ``rank_limit``: an upper bound for the minimal number of generators
- ``bound``: upper bound for the dimension (for considered being simple), default=0
OUTPUT:
A SimpleModulesGraph object. No computations are done after initialization.
Start the computation using the method ``compute()``.
"""
###########################
# basic parameters
###########################
self._level_limit = Integer(level_limit)
self._rank_limit = Integer(rank_limit)
self._signature = Integer(signature) % 8
self._weight = QQ(weight)
self._reduction = reduction
self._bound = bound
#########################################################
# Initialize the primes that need to be checked
# According to Theorem 4.21 in [BEF],
# in the worst case, we need to check primes p for which
# prime_pol_simple(p, weight) <= bound.
#########################################################
if primes is None:
p = 2
while True:
if prime_pol_simple(p, weight) > self._bound:
primes = list(prime_range(p))
break
else:
p = next_prime(p)
self._primes = primes
self._simple_color = colors.darkred.rgb(
) if simple_color is None else simple_color
self._nonsimple_color = colors.darkgreen.rgb(
) if nonsimple_color is None else nonsimple_color
self._vertex_colors = dict()
self._vertex_colors[self._simple_color] = list()
self._vertex_colors[self._nonsimple_color] = list()
self._heights = dict() # a height function for plotting
self._simple = list() # will contain the list of k-simple modules
super(SimpleModulesGraph, self).__init__()
def _rank(self, v):
r"""
Return a lower bound for the rank of the matrix U.
"""
self._U[self._U_rank] = v
K = self._U.base_ring()
if K is QQ:
return self._U.rank()
k = K.residue_field(self._good_prime)
try:
self._Ubar[self._U_rank] = v.change_ring(k)
except ValueError:
# Some denominator in v is not a unit mod q.
while True:
from sage.all import next_prime
self._good_prime = next_prime(self._good_prime.residue_field().characteristic())
self._good_prime = K.ideal(self._good_prime).factor()[0][0]
k = K.residue_field(self._good_prime)
try:
self._Ubar = self._U.change_ring(k)
except ValueError:
continue
if self._Ubar.rank() == self._U_rank + 1:
return self._U_rank + 1
return self._U_rank
def gen_instance(n, q, h, alpha=None, m=None, seed=None, s=None):
"""
Generate FHE-style LWE instance
:param n: dimension
:param q: modulus
:param alpha: noise rate (default: 8/q)
:param h: hamming weight of the secret (default: 2/3n)
:param m: number of samples (default: n)
"""
if seed is not None:
set_random_seed(seed)
q = next_prime(ceil(q) - 1, proof=False)
if alpha is None:
stddev = 3.2
else:
stddev = alpha * q / sqrt(2 * pi)
#RR = parent(alpha*q)
#stddev = alpha*q/RR(sqrt(2*pi))
if m is None:
m = n
K = GF(q, proof=False)
while 1:
A = random_matrix(K, m, n)
if A.rank() == n:
break
if s is not None:
c = A * s
else:
if h is None:
s = random_vector(ZZ, n, x=-1, y=1)
else:
S = [-1, 1]
s = [S[randint(0, 1)] for i in range(h)]
s += [0 for _ in range(n - h)]
shuffle(s)
s = vector(ZZ, s)
c = A * s
D = DiscreteGaussian(stddev)
for i in range(m):
c[i] += D()
u = random_vector(K, m)
print '(A, c) is n-dim LWE samples (with secret s) / (A, u) is uniform samples'
return A, c, u, s
def experiment(n=65,
q=next_prime(ceil(2**9)),
sigma=8,
m=178,
block_size=56,
tours=6):
alpha = sigma / q
L, e = gen_instance(n, alpha, q, m)
R, norms = run_instance(L, block_size, tours, e)
return L, R, e, norms
def max_coefficient_in_db(self):
r"""
Check how many coefficients we can generate from the eigenvalues in the database.
"""
from sage.all import next_prime
rec = self.get_db_record()
if rec is None:
return 0
prec_in_db = rec.get('prec')
return next_prime(prec_in_db) - 1
def max_coefficient_in_db(self):
r"""
Check how many coefficients we can generate from the eigenvalues in the database.
"""
from sage.all import next_prime
rec = self.get_db_record()
if rec is None:
return 0
prec_in_db = rec.get('prec')
return next_prime(prec_in_db)-1
def max_coefficient_in_db(self):
r"""
Check how many coefficients we can generate from the eigenvalues in the database.
"""
from sage.all import next_prime
prec_in_db = list(self._file_collection.find(self.key_dict(), ['prec']))
if len(prec_in_db) == 0:
return 0
max_prec_in_db = max(r['prec'] for r in prec_in_db)
return next_prime(max_prec_in_db)-1
def max_coefficient_in_db(self):
r"""
Check how many coefficients we can generate from the eigenvalues in the database.
"""
from sage.all import next_prime
prec_in_db = list(self._file_collection.find(self.key_dict(), ['prec']))
if len(prec_in_db) == 0:
return 0
max_prec_in_db = max(r['prec'] for r in prec_in_db)
return next_prime(max_prec_in_db)-1
def max_coefficient_in_db(self):
r"""
Check how many coefficients we can generate from the eigenvalues in the database.
"""
from sage.all import next_prime
recs = self._collection.find({'hecke_orbit_label':self.hecke_orbit_label})
if recs.count()==0:
return 0
prec_in_db = max(rec['prec'] for rec in recs)
return next_prime(prec_in_db)-1
def max_coefficient_in_db(self):
r"""
Check how many coefficients we can generate from the eigenvalues in the database.
"""
from sage.all import next_prime
recs = self._file_collection.find(self.key_dict())
if recs is None:
return 0
prec_in_db = max(rec['prec'] for rec in recs)
return next_prime(prec_in_db)-1
def max_coefficient_in_db(self):
r"""
Check how many coefficients we can generate from the eigenvalues in the database.
"""
from sage.all import next_prime
recs = self._collection.find(
{'hecke_orbit_label': self.hecke_orbit_label})
if recs.count() == 0:
return 0
prec_in_db = max(rec['prec'] for rec in recs)
return next_prime(prec_in_db) - 1
def __init__(self, surface):
if isinstance(surface, TranslationSurface):
base_ring = surface.base_ring()
from flatsurf.geometry.pyflatsurf_conversion import to_pyflatsurf
self._surface = to_pyflatsurf(surface)
else:
from flatsurf.geometry.pyflatsurf_conversion import sage_base_ring
base_ring, _ = sage_base_ring(surface)
self._surface = surface
# A model of the vector space R² in libflatsurf, e.g., to represent the
# vector associated to a saddle connection.
self.V2 = pyflatsurf.vector.Vectors(base_ring)
# We construct a spanning set of edges, that is a subset of the
# edges that form a basis of H_1(S, Sigma; Z)
# It comes together with a projection matrix
t, m = self._spanning_tree()
assert set(t.keys()) == set(f[0] for f in self._faces())
self.spanning_set = []
v = set(t.values())
for e in self._surface.edges():
if e.positive() not in v and e.negative() not in v:
self.spanning_set.append(e)
self.d = len(self.spanning_set)
assert 3*self.d - 3 == self._surface.size()
assert m.rank() == self.d
m = m.transpose()
# projection matrix from Z^E to H_1(S, Sigma; Z) in the basis
# of spanning edges
self.proj = matrix(ZZ, [r for r in m.rows() if not r.is_zero()])
self.Omega = self._intersection_matrix(t, self.spanning_set)
self.V = FreeModule(self.V2.base_ring(), self.d)
self.H = matrix(self.V2.base_ring(), self.d, 2)
for i in range(self.d):
s = self._surface.fromHalfEdge(self.spanning_set[i].positive())
self.H[i] = self.V2._isomorphic_vector_space(self.V2(s))
self.Hdual = self.Omega * self.H
# Note that we don't use Sage vector spaces because they are usually
# way too slow (in particular we avoid calling .echelonize())
self._U = matrix(self.V2._algebraic_ring(), self.d)
if self._U.base_ring() is not QQ:
from sage.all import next_prime
self._good_prime = self._U.base_ring().ideal(next_prime(2**30)).factor()[0][0]
self._Ubar = matrix(self._good_prime.residue_field(), self.d)
self._U_rank = 0
self.update_tangent_space_from_vector(self.H.transpose()[0])
self.update_tangent_space_from_vector(self.H.transpose()[1])
def dirichlet(R, euler_factors):
PS = PowerSeriesRing(R)
pef = zip(primes_first_n(len(euler_factors)), euler_factors)
an_list_bound = next_prime(pef[-1][0])
res = [1]*an_list_bound
for p, ef in pef:
k = RR(an_list_bound).log(p).floor()+1
foo = (1/PS(ef)).padded_list(k)
for i in range(1, k):
res[p**i] = foo[i]
extend_multiplicatively(res)
return res
def get_frob_list_HyperellipticCurve(f, h, bound=100):
C = HyperellipticCurve(f, h)
frob_list = []
p = 2
while p < bound:
try:
f = C.change_ring(GF(p)).frobenius_polynomial()
if f.degree() == 2 * C.genus():
frob_list.append([p, f.reverse()])
except:
pass
p = next_prime(p)
return frob_list
def __init__(self, signature=0, weight=2, level_limit=34, rank_limit=4, primes=None, simple_color=None, nonsimple_color=None,
reduction=True, bound=0):
"""
Initialize a SimpleModulesGraph containing finite quadratic modules of signature ``signature``.
They are checked for being ``weight``-simple if their minimal number of generators
is at most `rank_limit`.
INPUT:
- ``signature``: the signature of the modules
- ``weight``: check for cusp forms of weight ``weight``
- ``level_limit``: only check for anisotropic modules with level smaller than ``level_limit``
- ``rank_limit``: an upper bound for the minimal number of generators
- ``bound``: upper bound for the dimension (for considered being simple), default=0
OUTPUT:
A SimpleModulesGraph object. No computations are done after initialization.
Start the computation using the method ``compute()``.
"""
###########################
# basic parameters
###########################
self._level_limit = Integer(level_limit)
self._rank_limit = Integer(rank_limit)
self._signature = Integer(signature) % 8
self._weight = QQ(weight)
self._reduction = reduction
self._bound = bound
#########################################################
# Initialize the primes that need to be checked
# According to Theorem 4.21 in [BEF],
# in the worst case, we need to check primes p for which
# prime_pol_simple(p, weight) <= bound.
#########################################################
if primes is None:
p = 2
while True:
if prime_pol_simple(p, weight) > self._bound:
primes = list(prime_range(p))
break
else:
p = next_prime(p)
self._primes = primes
self._simple_color = colors.darkred.rgb() if simple_color is None else simple_color
self._nonsimple_color = colors.darkgreen.rgb() if nonsimple_color is None else nonsimple_color
self._vertex_colors = dict()
self._vertex_colors[self._simple_color] = list()
self._vertex_colors[self._nonsimple_color] = list()
self._heights = dict() # a height function for plotting
self._simple = list() # will contain the list of k-simple modules
super(SimpleModulesGraph, self).__init__()
def verify_curve(g, factorsRR_geom, bound = 300, RM_coeff = None):
D = ZZ(g.discriminant());
C = HyperellipticCurve(g)
p = 3;
ef = [];
while p < bound:
if D.mod(p) != ZZ(0):
ef += [C.change_ring(GF(p)).frobenius_polynomial().list()]
ef[-1].reverse()
p = next_prime(p)
rank_euler, _ = upperrank(ef)
rank_endo = rank_from_endo(factorsRR_geom)
if factorsRR_geom == ['RR', 'RR'] and RM_coeff is not None:
checkRM = RM_and_notCM(ef, RM_coeff)
return rank_endo == rank_euler and checkRM, rank_endo, rank_euler
else:
return rank_endo == rank_euler, rank_endo, rank_euler
def central_character_as_artin_rep(self):
"""
Returns the central character, i.e., determinant character
as a web character, but as an Artin representation
"""
if self.dimension() == 1:
return self
if 'central_character_as_artin_rep' in self._data:
return self._data['central_character_as_artin_rep']
return ArtinRepresentation(self._data['Dets'][self.galorbindex() - 1])
myfunc = self.central_char_function()
# Get the Artin field
nfgg = self.number_field_galois_group()
# Get its artin reps
arts = nfgg.ArtinReps()
# Filter for 1-dim
arts = [
a for a in arts
if ArtinRepresentation(str(a['Baselabel']) + "c1").dimension() == 1
]
artfull = [
ArtinRepresentation(str(a['Baselabel']) + "c" + str(a['GalConj']))
for a in arts
]
# hold = artfull
# Loop as we evaluate at primes until there is only one left
# Fix the return value to be what we want
artfull = [[a, a.central_char_function(), 2 * a.character_field()]
for a in artfull]
n = 2 * self.character_field()
p = 2
hard_primes = self.hard_primes()
while len(artfull) > 1:
if p not in hard_primes:
k = 0
while k < len(artfull):
if n * artfull[k][1](
p, artfull[k][2]) == artfull[k][2] * myfunc(p, n):
k += 1
else:
# Quick deletion of k-th term
artfull[k] = artfull[-1]
del artfull[-1]
p = next_prime(p)
self._data['central_character_as_artin_rep'] = artfull[0][0]
return artfull[0][0]
def gen_fhe_instance(n, q, alpha=None, h=None, m=None, seed=None):
"""
Generate FHE-style LWE instance
:param n: dimension
:param q: modulus
:param alpha: noise rate (default: 8/q)
:param h: hamming weight of the secret (default: 2/3n)
:param m: number of samples (default: n)
"""
if seed is not None:
set_random_seed(seed)
q = next_prime(ceil(q)-1, proof=False)
if alpha is None:
alpha = ZZ(8)/q
n, alpha, q = preprocess_params(n, alpha, q)
stddev = stddevf(alpha*q)
if m is None:
m = n
K = GF(q, proof=False)
A = random_matrix(K, m, n)
if h is None:
s = random_vector(ZZ, n, x=-1, y=1)
else:
S = [-1, 1]
s = [S[randint(0, 1)] for i in range(h)]
s += [0 for _ in range(n-h)]
shuffle(s)
s = vector(ZZ, s)
c = A*s
D = DiscreteGaussian(stddev)
for i in range(m):
c[i] += D()
return A, c
def gen_fhe_instance(n, q, alpha=None, h=None, m=None, seed=None):
"""
Generate FHE-style LWE instance
:param n: dimension
:param q: modulus
:param alpha: noise rate (default: 8/q)
:param h: hamming weight of the secret (default: 2/3n)
:param m: number of samples (default: n)
"""
if seed is not None:
set_random_seed(seed)
q = next_prime(ceil(q) - 1, proof=False)
if alpha is None:
alpha = ZZ(8) / q
n, alpha, q = preprocess_params(n, alpha, q)
stddev = stddevf(alpha * q)
if m is None:
m = n
K = GF(q, proof=False)
A = random_matrix(K, m, n)
if h is None:
s = random_vector(ZZ, n, x=-1, y=1)
else:
S = [-1, 1]
s = [S[randint(0, 1)] for i in range(h)]
s += [0 for _ in range(n - h)]
shuffle(s)
s = vector(ZZ, s)
c = A * s
D = DiscreteGaussian(stddev)
for i in range(m):
c[i] += D()
return A, c
def InsolublePlaces(f,h=0):
# List of primes at which the curve y²+h(x)*y=f(x) is not soluble
# Assumes f and h have integer coefficients
S = [] # List of primes at which not soluble
# Get eqn of the form y²=F(x)
F = f
if h:
F = 4*f + h**2
# Do we have a rational point an Infty ?
if F.degree()%2 or F.leading_coefficient().is_square():
return []
# Treat case of RR
if F.leading_coefficient() < 0 and F.number_of_real_roots() == 0:
S.append(0)
# Remove squares from lc(F)
d = F.degree()
lc = F.leading_coefficient()
t = F.variables()[0]
a = lc.squarefree_part() # lc/a is a square
a = lc//a
Zx = PolynomialRing(ZZ,'x')
F = Zx(a**(d-1) * F(t/a))
# Find primes of bad reduction for our model
D = F.disc()
P = Set(D.prime_divisors())
# Add primes at which Weil bounds do not guarantee a mod p point
g = (d-2)//2
Weil = []
p = 2
while (p+1)**2 <= 4 * g**2 * p:
Weil.append(p)
p = next_prime(p)
P = P.union(Set(Weil))
# Test for solubility
for p in P:
if not IsSolubleAt(F,p):
S.append(p)
S.sort()
return S
def InsolublePlaces(f, h=0):
# List of primes at which the curve y²+h(x)*y=f(x) is not soluble
# Assumes f and h have integer coefficients
S = [] # List of primes at which not soluble
# Get eqn of the form y²=F(x)
F = f
if h:
F = 4 * f + h**2
# Do we have a rational point an Infty ?
if F.degree() % 2 or F.leading_coefficient().is_square():
return []
# Treat case of RR
if F.leading_coefficient() < 0 and F.number_of_real_roots() == 0:
S.append(0)
# Remove squares from lc(F)
d = F.degree()
lc = F.leading_coefficient()
t = F.variables()[0]
a = lc.squarefree_part() # lc/a is a square
a = lc // a
Zx = PolynomialRing(ZZ, 'x')
F = Zx(a**(d - 1) * F(t / a))
# Find primes of bad reduction for our model
D = F.disc()
P = Set(D.prime_divisors())
# Add primes at which Weil bounds do not guarantee a mod p point
g = (d - 2) // 2
Weil = []
p = 2
while (p + 1)**2 <= 4 * g**2 * p:
Weil.append(p)
p = next_prime(p)
P = P.union(Set(Weil))
# Test for solubility
for p in P:
if not IsSolubleAt(F, p):
S.append(p)
S.sort()
return S
def genCurve(bits):
p = sg.next_prime(random.randint(2**(bits-1), 2**(bits)))
order = 0
while not(sg.is_prime(order)) or order == p or order == p+1:
A = random.randint(1, 10**5)
B = random.randint(1, 10**5)
if (4*A**3 + 27*B**2) % p != 0: ## Avoid singular curves ##
E = sg.EllipticCurve(sg.GF(p), [A,B])
order = E.cardinality()
P = E.random_point()
Q = E.random_point()
f = open("curves.csv", "a")
f.write(str(bits)+",")
f.write(str(A)+",")
f.write(str(B)+",")
f.write(str(p)+",")
f.write(str(order)+",")
f.write(str(P.xy()[0])+","+str(P.xy()[1])+",")
f.write(str(Q.xy()[0])+","+str(Q.xy()[1])+"\n")
f.close()
def genCurve(bits):
p = sg.next_prime(random.randint(2**(bits - 1), 2**(bits)))
order = 0
while not (sg.is_prime(order)) or order == p or order == p + 1:
A = random.randint(1, 10**5)
B = random.randint(1, 10**5)
if (4 * A**3 + 27 * B**2) % p != 0: ## Avoid singular curves ##
E = sg.EllipticCurve(sg.GF(p), [A, B])
order = E.cardinality()
P = E.random_point()
Q = E.random_point()
f = open("curves.csv", "a")
f.write(str(bits) + ",")
f.write(str(A) + ",")
f.write(str(B) + ",")
f.write(str(p) + ",")
f.write(str(order) + ",")
f.write(str(P.xy()[0]) + "," + str(P.xy()[1]) + ",")
f.write(str(Q.xy()[0]) + "," + str(Q.xy()[1]) + "\n")
f.close()
def central_character_as_artin_rep(self):
"""
Returns the central character, i.e., determinant character
as a web character, but as an Artin representation
"""
if self.dimension() == 1:
return self
if 'central_character_as_artin_rep' in self._data:
return self._data['central_character_as_artin_rep']
myfunc = self.central_char_function()
# Get the Artin field
nfgg = self.number_field_galois_group()
# Get its artin reps
arts = nfgg.ArtinReps()
# Filter for 1-dim
arts = [a for a in arts if ArtinRepresentation(str(a['Baselabel'])+"c1").dimension()==1]
artfull = [ArtinRepresentation(str(a['Baselabel'])+"c"+str(a['GalConj'])) for a in arts]
# hold = artfull
# Loop as we evaluate at primes until there is only one left
# Fix the return value to be what we want
artfull = [[a, a.central_char_function(),2*a.character_field()] for a in artfull]
n = 2*self.character_field()
p = 2
disc = nfgg.discriminant()
while len(artfull)>1:
if (disc % p) != 0:
k=0
while k<len(artfull):
if n*artfull[k][1](p,artfull[k][2]) == artfull[k][2]*myfunc(p,n):
k += 1
else:
# Quick deletion of k-th term
artfull[k] = artfull[-1]
del artfull[-1]
p = next_prime(p)
self._data['central_character_as_artin_rep'] = artfull[0][0]
return artfull[0][0]
xs.append(1)
xs = vector(xs)
A = []
for i in range(m):
cs = [randrange(hi) for _ in range(n-1)]
cs.append(-sum((x*c) for x, c in zip(xs, cs)))
assert sum(c*x for c, x in zip(cs, xs)) == 0
A.append(cs)
assert xs == small_lgs(matrix(A))
print ' Testing homogenous LGS modulo prime'
hi = 2**128
lo = 2**100
p = next_prime(hi)
n = 40 # num vars
m = 35 # num equations
xs = [randrange(lo) for _ in range(n)]
xs = vector(xs)
A = []
for i in range(m):
cs = [randrange(p) for _ in range(n-1)]
last_c = -sum(c*x for c, x in zip(cs, xs)) * inverse_mod(xs[-1], p)
cs.append(last_c % p)
assert sum(c*x for c, x in zip(cs, xs))%p == 0
A.append(cs)
A = matrix(ZZ, A)
def render_newform_webpage(label):
try:
newform = WebNewform.by_label(label)
except (KeyError, ValueError) as err:
return abort(404, err.args)
info = to_dict(request.args)
info['format'] = info.get('format',
'embed' if newform.dim > 1 else 'satake')
p, maxp = 2, 10
if info['format'] in ['satake', 'satake_angle']:
while p <= maxp:
if newform.level % p == 0:
maxp = next_prime(maxp)
p = next_prime(p)
errs = []
info['n'] = info.get('n', '2-%s' % maxp)
try:
info['CC_n'] = integer_options(info['n'], 1000)
except (ValueError, TypeError) as err:
info['n'] = '2-%s' % maxp
info['CC_n'] = range(2, maxp + 1)
if err.args and err.args[0] == 'Too many options':
errs.append(r"Only \(a_n\) up to %s are available" %
(newform.cqexp_prec - 1))
else:
errs.append(
"<span style='color:black'>n</span> must be an integer, range of integers or comma separated list of integers"
)
maxm = min(newform.dim, 20)
info['m'] = info.get('m', '1-%s' % maxm)
try:
info['CC_m'] = integer_options(info['m'], 1000)
except (ValueError, TypeError) as err:
info['m'] = '1-%s' % maxm
info['CC_m'] = range(1, maxm + 1)
if err.args and err.args[0] == 'Too many options':
errs.append(
'Web interface only supports 1000 embeddings at a time. Use download link to get more (may take some time).'
)
else:
errs.append(
"<span style='color:black'>Embeddings</span> must be an integer, range of integers or comma separated list of integers"
)
try:
info['prec'] = int(info.get('prec', 6))
if info['prec'] < 1 or info['prec'] > 15:
raise ValueError
except (ValueError, TypeError):
info['prec'] = 6
errs.append(
"<span style='color:black'>Precision</span> must be a positive integer, at most 15 (for higher precision, use the download button)"
)
newform.setup_cc_data(info)
if newform.cqexp_prec != 0 and max(info['CC_n']) >= newform.cqexp_prec:
errs.append(r"Only \(a_n\) up to %s are available" %
(newform.cqexp_prec - 1))
if errs:
flash(Markup("<br>".join(errs)), "error")
return render_template("cmf_newform.html",
info=info,
newform=newform,
properties2=newform.properties,
downloads=newform.downloads,
credit=credit(),
bread=newform.bread,
learnmore=learnmore_list(),
title=newform.title,
friends=newform.friends)
def DirichletCoefficients(euler_factors, bound = None):
"""
INPUT:
- ``euler_factors`` -- a dictionary p --> Lp.list() where Lp \in 1 + ZZ[t]
OUTPUT:
A list of length bound + 1 -- [0, a1, a2, ... , a_bound]
EXAMPLES:
sage: E17a2_euler_factors = {2: [1, 1, 2], 3: [1, 0, 3], 5: [1, 2, 5], 7: [1, -4, 7], 11: [1, 0, 11], 13: [1, 2, 13], 17: [1, -1], 19: [1, 4, 19], 23: [1, -4, 23], 29: [1, -6, 29], 31: [1, -4, 31], 37: [1, 2, 37], 41: [1, 6, 41], 43: [1, -4, 43], 47: [1, 0, 47], 53: [1, -6, 53], 59: [1, 12, 59], 61: [1, 10, 61], 67: [1, -4, 67], 71: [1, 4, 71], 73: [1, 6, 73], 79: [1, -12, 79], 83: [1, 4, 83], 89: [1, -10, 89], 97: [1, -2, 97]}
sage: A = DirichletCoefficients(E17a2_euler_factors, bound = 99)
sage: A == [0, 1, -1, 0, -1, -2, 0, 4, 3, -3, 2, 0, 0, -2, -4, 0, -1, 1, 3, -4, 2, 0, 0, 4, 0, -1, 2, 0, -4, 6, 0, 4, -5, 0, -1, -8, 3, -2, 4, 0, -6, -6, 0, 4, 0, 6, -4, 0, 0, 9, 1, 0, 2, 6, 0, 0, 12, 0, -6, -12, 0, -10, -4, -12, 7, 4, 0, 4, -1, 0, 8, -4, -9, -6, 2, 0, 4, 0, 0, 12, 2, 9, 6, -4, 0, -2, -4, 0, 0, 10, -6, -8, -4, 0, 0, 8, 0, 2, -9, 0]
True
sage: E33a3_euler_factors = {2: [1, -1, 2], 3: [1, 1], 5: [1, 2, 5], 7: [1, -4, 7], 11: [1, -1], 13: [1, 2, 13], 17: [1, 2, 17], 19: [1, 0, 19], 23: [1, -8, 23], 29: [1, 6, 29], 31: [1, 8, 31], 37: [1, -6, 37], 41: [1, 2, 41], 43: [1, 0, 43], 47: [1, -8, 47], 53: [1, -6, 53], 59: [1, 4, 59], 61: [1, -6, 61], 67: [1, 4, 67], 71: [1, 0, 71], 73: [1, 14, 73], 79: [1, 4, 79], 83: [1, -12, 83], 89: [1, 6, 89], 97: [1, -2, 97]}
sage: A = DirichletCoefficients(E33a3_euler_factors, bound = 99)
sage: A == [0, 1, 1, -1, -1, -2, -1, 4, -3, 1, -2, 1, 1, -2, 4, 2, -1, -2, 1, 0, 2, -4, 1, 8, 3, -1, -2, -1, -4, -6, 2, -8, 5, -1, -2, -8, -1, 6, 0, 2, 6, -2, -4, 0, -1, -2, 8, 8, 1, 9, -1, 2, 2, 6, -1, -2, -12, 0, -6, -4, -2, 6, -8, 4, 7, 4, -1, -4, 2, -8, -8, 0, -3, -14, 6, 1, 0, 4, 2, -4, 2, 1, -2, 12, 4, 4, 0, 6, -3, -6, -2, -8, -8, 8, 8, 0, -5, 2, 9, 1]
True
"""
if bound is None:
bound = next_prime( max(euler_factors.keys() ) ) - 1;
degree = 0;
for p, L in euler_factors.iteritems():
if len(L) > degree + 1:
degree = len(L) - 1;
R = PolynomialRing(ZZ, degree, "a")
S = PowerSeriesRing(R, default_prec = ceil(log(bound)/log(2)));
P = [1] + list(R.gens());
recursion = (1/S(P)).list();
A = [None]*(bound + 1);
A[0] = 0;
A[1] = 1;
i = 1;
# sieve through [1, bound]
while i <= bound:
if A[i] is None:
#i is a prime
if i in euler_factors:
euler_i = euler_factors[i][1:];
else:
euler_i = [];
euler_i += [0] * (degree - len(euler_i)); # fill with zeros if necessary
# deal with its powers
r = 1;
ipower = i # i^r
while ipower <= bound:
A[ipower] = recursion[r](euler_i)
ipower *= i;
r += 1;
# deal with multiples of its powers by smaller numbers
for m in range(2, bound):
if A[m] is not None and m%i != 0:
ipower = i
while m*ipower <= bound:
assert A[m*ipower] is None
A[m*ipower] = A[m] * A[ipower]
ipower *= i
i += 1
return A;
def _compute_simple_modules_graph_from_startpoint(self, s, p=None, cut_nonsimple_aniso=True, fast=1):
# for forking this is necessary
logger = get_logger(s)
# print logger
k = self._weight
###########################################################
# Determine which primes need to be checked
# According to the proof of Proposition XX in [BEF], we
# only need to check primesnot dividing the 6*level(s),
# for which prime_pol(s,p,k) <= 0.
# For those primes, we check if there is any
# k-simple fqm in s.C(p) and if not, we do not have to
# consider p anymore.
###########################################################
if p == None:
p = 2
N = Integer(6) * s.level()
slp = N.prime_factors()
for q in prime_range(next_prime(N) + 1):
if not q in slp:
logger.info(
"Smallest prime not dividing 6*level({0}) = {1} is p = {2}".format(s, Integer(6) * s.level(), q))
p = q
break
while prime_pol(s, p, k) <= 0 or p in slp:
p = next_prime(p)
p = uniq(prime_range(p) + slp)
logger.info("Starting with s = {0} and primes = {1}".format(s, p))
if isinstance(p, list):
primes = p
else:
primes = [p]
simple = s.is_simple(k, reduction = self._reduction, bound = self._bound)
if not simple:
logger.info("{0} is not simple.".format(s))
# print simple
s = FQM_vertex(s)
# print s
self.add_vertex(s)
# print "added vertex ", s
############################################################
# Starting from the list of primes we generated,
# we now compute which primes we actually need to consider.
# That is, the primes such that there is a fqm in s.C(p)
# which is k-simple.
############################################################
np = list()
if cut_nonsimple_aniso and simple:
for i in range(len(primes)):
p = primes[i]
fs = False
for t in s.genus_symbol().C(p, False):
if t.is_simple(k, bound = self._bound):
fs = True
logger.debug("Setting fs = True")
break
if fs:
np.append(p)
primes = np
# print "here", primes
logger.info("primes for graph for {0}: {1}".format(s, primes))
# if len(primes) == 0:
# return
heights = self._heights
h = 0
if not heights.has_key(h):
heights[h] = [s]
else:
if heights[h].count(s) == 0:
heights[h].append(s)
vertex_colors = self._vertex_colors
nonsimple_color = self._nonsimple_color
simple_color = self._simple_color
# Bs contains the modules of the current level (= height = h)
Bs = [s]
# set the correct color for the vertex s
if simple:
if vertex_colors[simple_color].count(s) == 0:
vertex_colors[simple_color].append(s)
elif vertex_colors[nonsimple_color].count(s) == 0:
vertex_colors[nonsimple_color].append(s)
###################################################
# MAIN LOOP
# we loop until we haven't found any simple fqm's
###################################################
while simple:
h = h + 1
if not heights.has_key(h):
heights[h] = list()
# the list Bss will contain the k-simple modules of the next height level
# recall that Bs contains the modules of the current height level
Bss = list()
simple = False
# checklist = modules that we need to check for with .is_simple(k)
# we assemble this list because afterwards
# we check them in parallel
checklist = []
for s1 in Bs:
Bs2 = list()
for p in primes:
# check if we really need to check p for s1
# otherwise none of the fqm's in s1.C(p) are simple
# and we will not consider them.
if prime_pol(s1.genus_symbol(), p, k) <= 0:
Bs2 = Bs2 + s1.genus_symbol().C(p, False)
else:
logger.info(
"Skipping p = {0} for s1 = {1}".format(p, s1))
# print "Skipping p = {0} for s1 = {1}".format(p, s1)
# print "Bs2 = ", Bs2
# now we actually check the symbols in Bs2
for s2 in Bs2:
if s2.max_rank() > self._rank_limit:
# we skip s2 if its minimal number of generators
# is > than the given rank_limit.
continue
s2 = FQM_vertex(s2)
skip = False
for v in self._heights[h]:
# we skip symbols that correspond to isomorphic modules
if v.genus_symbol().defines_isomorphic_module(s2.genus_symbol()):
skip = True
logger.debug(
"skipping {0} b/c isomorphic to {1}".format(s2.genus_symbol(), v.genus_symbol()))
s2 = v
break
if skip:
continue
if not skip:
self.add_vertex(s2)
heights[h].append(s2)
self.update_edges(s2, h, fast=fast)
# before using the actual dimension formula
# we check if there is already a non-k-simple neighbor
# (an incoming edge from a non-k-simple module)
# which would imply that s2 is not k-simple.
has_nonsimple_neighbor = False
for e in self.incoming_edges(s2):
if vertex_colors[nonsimple_color].count(e[0]) > 0:
has_nonsimple_neighbor = True
logger.debug(
"Has nonsimple neighbor: {0}".format(s2.genus_symbol()))
break
if has_nonsimple_neighbor:
#not simple
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
else:
checklist.append((s2, k, self._reduction, self._bound))
logger.debug("checklist = {0}".format(checklist))
# check the modules in checklist
# for being k-simple
# this is done in parallel
# when a process returns
# we add the vertex and give it its appropriate color
if NCPUS1 == 1:
checks = [([[s[0]]],check_simple(*s)) for s in checklist]
else:
checks = list(check_simple(checklist))
logger.info("checks = {0}".format(checks))
for check in checks:
s2 = check[0][0][0]
if check[1]:
simple = True
logger.info(
"Found simple module: {0}".format(s2.genus_symbol()))
Bss.append(s2)
if not vertex_colors[simple_color].count(s2) > 0:
vertex_colors[simple_color].append(s2)
else:
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
Bs = Bss
simple = [v.genus_symbol() for v in vertex_colors[simple_color]]
self._simple = uniq(simple)
Z = [0,]*len(X)
for i in range(len(X)):
x = X[i]
z = previous_prime(x) # Returns z < x
if is_prime(x): # galois.prev_prime() returns z <= x
z = x
Z[i] = int(z)
d = {"X": X, "Z": Z}
save_pickle(d, FOLDER, "prev_prime.pkl")
set_seed(SEED + 304)
X = [random.randint(1, 100) for _ in range(20)] + [random.randint(100, 1_000_000) for _ in range(20)]
Z = [0,]*len(X)
for i in range(len(X)):
x = X[i]
z = next_prime(x)
Z[i] = int(z)
d = {"X": X, "Z": Z}
save_pickle(d, FOLDER, "next_prime.pkl")
set_seed(SEED + 305)
X = [random.randint(-100, 100) for _ in range(20)] + [random.randint(100, 1_000_000_000) for _ in range(20)]
Z = [0,]*len(X)
for i in range(len(X)):
x = X[i]
z = is_prime(x)
Z[i] = bool(z)
d = {"X": X, "Z": Z}
save_pickle(d, FOLDER, "is_prime.pkl")
# set_seed(SEED + 306)
def _compute_simple_modules_graph_from_startpoint(self,
s,
p=None,
cut_nonsimple_aniso=True,
fast=1):
# for forking this is necessary
logger = get_logger(s)
# print logger
k = self._weight
###########################################################
# Determine which primes need to be checked
# According to the proof of Proposition XX in [BEF], we
# only need to check primesnot dividing the 6*level(s),
# for which prime_pol(s,p,k) <= 0.
# For those primes, we check if there is any
# k-simple fqm in s.C(p) and if not, we do not have to
# consider p anymore.
###########################################################
if p == None:
p = 2
N = Integer(6) * s.level()
slp = N.prime_factors()
for q in prime_range(next_prime(N) + 1):
if not q in slp:
logger.info(
"Smallest prime not dividing 6*level({0}) = {1} is p = {2}"
.format(s,
Integer(6) * s.level(), q))
p = q
break
while prime_pol(s, p, k) <= 0 or p in slp:
p = next_prime(p)
p = uniq(prime_range(p) + slp)
logger.info("Starting with s = {0} and primes = {1}".format(s, p))
if isinstance(p, list):
primes = p
else:
primes = [p]
simple = s.is_simple(k, reduction=self._reduction, bound=self._bound)
if not simple:
logger.info("{0} is not simple.".format(s))
# print simple
s = FQM_vertex(s)
# print s
self.add_vertex(s)
# print "added vertex ", s
############################################################
# Starting from the list of primes we generated,
# we now compute which primes we actually need to consider.
# That is, the primes such that there is a fqm in s.C(p)
# which is k-simple.
############################################################
np = list()
if cut_nonsimple_aniso and simple:
for i in range(len(primes)):
p = primes[i]
fs = False
for t in s.genus_symbol().C(p, False):
if t.is_simple(k, bound=self._bound):
fs = True
logger.debug("Setting fs = True")
break
if fs:
np.append(p)
primes = np
# print "here", primes
logger.info("primes for graph for {0}: {1}".format(s, primes))
# if len(primes) == 0:
# return
heights = self._heights
h = 0
if not heights.has_key(h):
heights[h] = [s]
else:
if heights[h].count(s) == 0:
heights[h].append(s)
vertex_colors = self._vertex_colors
nonsimple_color = self._nonsimple_color
simple_color = self._simple_color
# Bs contains the modules of the current level (= height = h)
Bs = [s]
# set the correct color for the vertex s
if simple:
if vertex_colors[simple_color].count(s) == 0:
vertex_colors[simple_color].append(s)
elif vertex_colors[nonsimple_color].count(s) == 0:
vertex_colors[nonsimple_color].append(s)
###################################################
# MAIN LOOP
# we loop until we haven't found any simple fqm's
###################################################
while simple:
h = h + 1
if not heights.has_key(h):
heights[h] = list()
# the list Bss will contain the k-simple modules of the next height level
# recall that Bs contains the modules of the current height level
Bss = list()
simple = False
# checklist = modules that we need to check for with .is_simple(k)
# we assemble this list because afterwards
# we check them in parallel
checklist = []
for s1 in Bs:
Bs2 = list()
for p in primes:
# check if we really need to check p for s1
# otherwise none of the fqm's in s1.C(p) are simple
# and we will not consider them.
if prime_pol(s1.genus_symbol(), p, k) <= 0:
Bs2 = Bs2 + s1.genus_symbol().C(p, False)
else:
logger.info("Skipping p = {0} for s1 = {1}".format(
p, s1))
# print "Skipping p = {0} for s1 = {1}".format(p, s1)
# print "Bs2 = ", Bs2
# now we actually check the symbols in Bs2
for s2 in Bs2:
if s2.max_rank() > self._rank_limit:
# we skip s2 if its minimal number of generators
# is > than the given rank_limit.
continue
s2 = FQM_vertex(s2)
skip = False
for v in self._heights[h]:
# we skip symbols that correspond to isomorphic modules
if v.genus_symbol().defines_isomorphic_module(
s2.genus_symbol()):
skip = True
logger.debug(
"skipping {0} b/c isomorphic to {1}".format(
s2.genus_symbol(), v.genus_symbol()))
s2 = v
break
if skip:
continue
if not skip:
self.add_vertex(s2)
heights[h].append(s2)
self.update_edges(s2, h, fast=fast)
# before using the actual dimension formula
# we check if there is already a non-k-simple neighbor
# (an incoming edge from a non-k-simple module)
# which would imply that s2 is not k-simple.
has_nonsimple_neighbor = False
for e in self.incoming_edges(s2):
if vertex_colors[nonsimple_color].count(e[0]) > 0:
has_nonsimple_neighbor = True
logger.debug("Has nonsimple neighbor: {0}".format(
s2.genus_symbol()))
break
if has_nonsimple_neighbor:
#not simple
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
else:
checklist.append((s2, k, self._reduction, self._bound))
logger.debug("checklist = {0}".format(checklist))
# check the modules in checklist
# for being k-simple
# this is done in parallel
# when a process returns
# we add the vertex and give it its appropriate color
if NCPUS1 == 1:
checks = [([[s[0]]], check_simple(*s)) for s in checklist]
else:
checks = list(check_simple(checklist))
logger.info("checks = {0}".format(checks))
for check in checks:
s2 = check[0][0][0]
if check[1]:
simple = True
logger.info("Found simple module: {0}".format(
s2.genus_symbol()))
Bss.append(s2)
if not vertex_colors[simple_color].count(s2) > 0:
vertex_colors[simple_color].append(s2)
else:
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
Bs = Bss
simple = [v.genus_symbol() for v in vertex_colors[simple_color]]
self._simple = uniq(simple)
def get_type_1_primes(K, aux_prime_count=3, loop_curves=False):
"""Compute the type 1 primes"""
C_K = K.class_group()
aux_primes = [Q_2]
prime_to_append = Q_2
for _ in range(1, aux_prime_count):
prime_to_append = next_prime(prime_to_append)
aux_primes.append(prime_to_append)
# We need at least one aux prime that splits to have provable termination
my_legendre_symbols = set([legendre_symbol(K._D, p) for p in aux_primes])
if 1 not in my_legendre_symbols:
prime_to_append = next_prime(prime_to_append)
while legendre_symbol(K._D, prime_to_append) != 1:
prime_to_append = next_prime(prime_to_append)
aux_primes.append(prime_to_append)
D_dict = {}
for q in aux_primes:
frak_q = K.primes_above(q)[0]
residue_field = frak_q.residue_field(names="z")
residue_field_card = residue_field.cardinality()
frak_q_class_group_order = C_K(frak_q).multiplicative_order()
exponent = 12 * frak_q_class_group_order
running_D = q
if loop_curves:
weil_polys = get_weil_polys(residue_field)
else:
weil_polys = R.weil_polynomials(2, residue_field_card)
# First add the C((0,0), frak_q) terms
for wp in weil_polys:
D = get_C00(wp, residue_field_card, exponent)
D = Integer(D)
# If q splits then we know C((0,0),frak_q) is non-zero
# so we can insist on only adding D if it is non-zero
if legendre_symbol(K._D, q) == 1:
if D != 0:
running_D = lcm(running_D, D)
else:
# Here we do not provably know that the integer is non-zero
# so add it directly to the running_D
running_D = lcm(running_D, D)
# Put the other term in the lcm before adding to dictionary
running_D = lcm(residue_field_card ** exponent - 1, running_D)
D_dict[q] = running_D
output = gcd(list(D_dict.values()))
output = set(output.prime_divisors())
# Add the bad formal immersion primes
output = output.union(set(prime_range(62)), {71})
# Sort and return
output = list(output)
output.sort()
return output
import sys
from random_curve_iterator import RandomCurveIterator
from sage.all import GF, next_prime
if __name__ == "__main__":
p = next_prime(100)
F = GF(p)
count = 0
for ec in RandomCurveIterator(F):
if ec.cardinality() == p - 1:
print(ec)
sys.exit()
nfcc = [int(z.order()) for z in nfcc]
nfcc = lcm(nfcc)
if not cfz.divides(nfcc):
print("Failure " + str(cfz) + " divides " + str(nfcc) + " from " + label)
sys.exit()
R, x = QQ['x'].objgen()
pol1 = R.cyclotomic_polynomial(nfcc)
K, z = NumberField(R.cyclotomic_polynomial(nfcc), 'z').objgen()
RR, y = K['y'].objgen()
zsmall = z**(nfcc / cfz)
allpols = [
sum(y**k * sum(pp[k][j] * zsmall**j for j in range(len(pp[k])))
for k in range(len(pp))) for pp in ar.local_factors_table()
]
allroots = [myroots(pp, nfcc, z) for pp in allpols]
outfile.write(str(nfcc) + "\n")
outfile.write(str(allroots) + "\n")
j = 0
p = 1
while j < bound:
p = next_prime(p)
outfile.write(str(ar.any_prime_to_cc_index(p)) + "\n")
j += 1
#plist = [ar.any_prime_to_cc_index(p) for p in primes_first_n(bound)]
#for j in plist:
# outfile.write(str(j)+"\n")
outfile.close()
cfz = ZZ(cf)
nf = ar.nf()
nfcc = nf.conjugacy_classes()
nfcc = [int(z.order()) for z in nfcc]
nfcc = lcm(nfcc)
if not cfz.divides(nfcc):
print "Failure "+str(cfz)+" divides "+str(nfcc)+" from "+label
sys.exit()
R,x = QQ['x'].objgen()
pol1 = R.cyclotomic_polynomial(nfcc)
K,z=NumberField(R.cyclotomic_polynomial(nfcc),'z').objgen()
RR,y = K['y'].objgen()
zsmall = z**(nfcc/cfz)
allpols = [sum(y**k * sum(pp[k][j] * zsmall**j for j in range(len(pp[k]))) for k in range(len(pp))) for pp in ar.local_factors_table()]
allroots = [myroots(pp, nfcc, z) for pp in allpols]
outfile.write(str(nfcc)+"\n")
outfile.write(str(allroots)+"\n")
j=0
p=1
while j<bound:
p = next_prime(p)
outfile.write(str(ar.any_prime_to_cc_index(p))+"\n")
j+=1
#plist = [ar.any_prime_to_cc_index(p) for p in primes_first_n(bound)]
#for j in plist:
# outfile.write(str(j)+"\n")
outfile.close()