def to_polredabs(K):
"""
INPUT:
* "K" - a number field
OUTPUT:
* "phi" - an isomorphism K -> L, where L = QQ['x']/f and f a polynomial such that f = polredabs(f)
"""
R = PolynomialRing(QQ,'x')
x = R.gen(0)
if K == QQ:
L = QQ.extension(x,'w')
return QQ.hom(L)
L = K.absolute_field('a')
m1 = L.structure()[1]
f = L.absolute_polynomial()
g = pari(f).polredabs(1)
g,h = g[0].sage(locals={'x':x}),g[1].lift().sage(locals={'x':x})
if debug:
print 'f',f
print 'g',g
print 'h',h
M = QQ.extension(g,'w')
m2 = L.hom([h(M.gen(0))])
return m2*m1
def to_polredabs(K):
"""
INPUT:
* "K" - a number field
OUTPUT:
* "phi" - an isomorphism K -> L, where L = QQ['x']/f and f a polynomial such that f = polredabs(f)
"""
R = PolynomialRing(QQ, 'x')
x = R.gen(0)
if K == QQ:
L = QQ.extension(x, 'w')
return QQ.hom(L)
L = K.absolute_field('a')
m1 = L.structure()[1]
f = L.absolute_polynomial()
g = pari(f).polredabs(1)
g, h = g[0].sage(locals={'x': x}), g[1].lift().sage(locals={'x': x})
if debug:
print('f', f)
print('g', g)
print('h', h)
M = QQ.extension(g, 'w')
m2 = L.hom([h(M.gen(0))])
return m2 * m1
def EllipticCurve_from_hoeij_data(line):
"""Given a line of the file "http://www.math.fsu.edu/~hoeij/files/X1N/LowDegreePlaces"
that is actually corresponding to an elliptic curve, this function returns the elliptic
curve corresponding to this
"""
Rx=PolynomialRing(QQ,'x')
x = Rx.gen(0)
Rxy = PolynomialRing(Rx,'y')
y = Rxy.gen(0)
N=ZZ(line.split(",")[0].split()[-1])
x_rel=Rx(line.split(',')[-2][2:-4])
assert x_rel.leading_coefficient()==1
y_rel=line.split(',')[-1][1:-5]
K = QQ.extension(x_rel,'x')
x = K.gen(0)
y_rel=Rxy(y_rel).change_ring(K)
y_rel=y_rel/y_rel.leading_coefficient()
if y_rel.degree()==1:
y = - y_rel[0]
else:
#print "needing an extension!!!!"
L = K.extension(y_rel,'y')
y = L.gen(0)
K = L
#B=L.absolute_field('s')
#f1,f2 = B.structure()
#x,y=f2(x),f2(y)
r = (x**2*y-x*y+y-1)/x/(x*y-1)
s = (x*y-y+1)/x/y
b = r*s*(r-1)
c = s*(r-1)
E=EllipticCurve([1-c,-b,-b,0,0])
return N,E,K
def isogeny_primes(coeffs, **kwargs):
del kwargs["label"]
f = R(coeffs)
K = QQ.extension(f, "a")
kwargs = {**LMFDB_DEFAULTS, **kwargs}
primes, _ = get_isogeny_primes(K, **kwargs)
return sorted(EC_Q_ISOGENY_PRIMES) + sorted(primes.difference(EC_Q_ISOGENY_PRIMES))
def EllipticCurve_from_hoeij_data(line):
"""Given a line of the file "http://www.math.fsu.edu/~hoeij/files/X1N/LowDegreePlaces"
that is actually corresponding to an elliptic curve, this function returns the elliptic
curve corresponding to this
"""
Rx = PolynomialRing(QQ, 'x')
x = Rx.gen(0)
Rxy = PolynomialRing(Rx, 'y')
y = Rxy.gen(0)
N = ZZ(line.split(",")[0].split()[-1])
x_rel = Rx(line.split(',')[-2][2:-4])
assert x_rel.leading_coefficient() == 1
y_rel = line.split(',')[-1][1:-5]
K = QQ.extension(x_rel, 'x')
x = K.gen(0)
y_rel = Rxy(y_rel).change_ring(K)
y_rel = y_rel / y_rel.leading_coefficient()
if y_rel.degree() == 1:
y = -y_rel[0]
else:
#print("needing an extension!!!!")
L = K.extension(y_rel, 'y')
y = L.gen(0)
K = L
#B=L.absolute_field('s')
#f1,f2 = B.structure()
#x,y=f2(x),f2(y)
r = (x**2 * y - x * y + y - 1) / x / (x * y - 1)
s = (x * y - y + 1) / x / y
b = r * s * (r - 1)
c = s * (r - 1)
E = EllipticCurve([1 - c, -b, -b, 0, 0])
return N, E, K
def test_semi_stable_frobenius_polynomial_t():
# an example where the frobenius polynomial depends on the purely ramified extension
# we make
x = polygen(QQ)
K = QQ.extension(x - 1, "one")
E = EllipticCurve(K, [49, 343])
assert E.discriminant() == -(2**4) * 31 * 7**6
assert E.j_invariant() == K(2**8 * 3**3) / 31
f1 = semi_stable_frobenius_polynomial(E, K * 7, 1)
f2 = semi_stable_frobenius_polynomial(E, K * 7, -1)(x=-x)
assert f1 == f2
def test_rational_isogeny_primes():
x = polygen(QQ)
K = QQ.extension(x - 1, "D")
superset, _ = get_isogeny_primes(K, **TEST_SETTINGS)
improperly_ruled_out = EC_Q_ISOGENY_PRIMES.difference(superset)
assert improperly_ruled_out == set()
todo = set(superset).difference(EC_Q_ISOGENY_PRIMES)
# would be nice if we could automatically do QQ
# i.e. we could rule out 23 as well.
assert todo == set([23])
def qexp_as_nf_elt(self, prec=None):
assert self.has_exact_qexp
if prec is None:
qexp = self.qexp
else:
qexp = self.qexp[:prec + 1]
if self.dim == 1:
return [QQ(i[0]) for i in self.qexp]
R = PolynomialRing(QQ, 'x')
K = QQ.extension(R(self.field_poly), 'a')
if self.hecke_ring_power_basis:
return [K(c) for c in qexp]
else:
# need to add code to hande cyclotomic_generators
assert self.hecke_ring_numerators, self.hecke_ring_denominators
basis_data = zip(self.hecke_ring_numerators,
self.hecke_ring_denominators)
betas = [K([ZZ(c) / den for c in num]) for num, den in basis_data]
return [
sum(c * beta for c, beta in zip(coeffs, betas)) for coeffs in qexp
]
def test_get_isogeny_primes(coeffs):
f = R(coeffs)
K = QQ.extension(f, "a")
_, _ = get_isogeny_primes(K, **TEST_SETTINGS)
