def _torsion_poly(ell, P=None):
"""
Computes the ell-th gauss period. If `P` is given, it must be a
polynomial ring into which the result is coerced.
This is my favourite equality:
sage: all(_torsion_poly(n)(I) == I^n*lucas_number2(n,1,-1) for n in range(1,10))
True
"""
if P is None:
P, R, = PolynomialRing(ZZ, 'x'), ZZ,
elif P.characteristic() == 0:
R = ZZ
else:
R = Zp(P.characteristic(), prec=1, type='capped-rel')
t = [1, 0]
for k in range(1, ell/2 + 1):
m = R(ell - 2*k + 2) * R(ell - 2*k + 1) / (R(ell - k) * R(k))
t.append(-t[-2] * m)
t.append(0)
return P(list(reversed(t))).shift(ell % 2 - 1)
def _torsion_poly(ell, P=None):
"""
Computes the ell-th gauss period. If `P` is given, it must be a
polynomial ring into which te result is coerced.
This is my favourite equality:
sage: all(_torsion_poly(n)(I) == I^n*lucas_number2(n,1,-1) for n in range(1,10))
True
"""
if P is None:
P, R, = PolynomialRing(ZZ, 'x'), ZZ,
elif P.characteristic() == 0:
R = ZZ
else:
R = Zp(P.characteristic(), prec=1, type='capped-rel')
t = [1, 0]
for k in range(1, ell/2 + 1):
m = R(ell - 2*k + 2) * R(ell - 2*k + 1) / (R(ell - k) * R(k))
t.append(-t[-2] * m)
t.append(0)
return P(list(reversed(t))).shift(ell % 2 - 1)
