def eta_p(q):
'''
q: an instance of jordan_block.JordanBlocks.
'''
p = q.p
hasse_inv = q.hasse_invariant__OMeara()
detb = q.Gram_det()
n = q.dim()
h1 = hilbert_symbol(detb, detb * (-1) ** ((n - 1) // 2), p)
h2 = hilbert_symbol(-1, -1, p)
return hasse_inv * h1 * h2 ** ((n ** 2 - 1) // 8)
def _invariants_2_even(b1, q2):
n = q2.dim() + 2
m = b1.m
q = b1 + q2
xi = xi_p(q)
xi_dash = xi_to_xi_dash(xi)
xi_hat = xi_p(q2)
xi_hat_dash = xi_to_xi_dash(xi_hat)
# Definition of eta_tilde
if b1.type == 'u' and small_d(q2) % 2 == 0:
_q = JordanBlocks([(m, b1._mat_prim[(1, 1)])], two)
eta_tilde = eta_p(_q + q2)
elif b1.type != 'u' and xi_hat != 0:
eta_tilde = ((-1) ** (((n - 1) ** 2 - 1) // 8) * q2.hasse_invariant__OMeara()
* hilbert_symbol(two ** m,
(-1) ** (n // 2 - 1) * q2.Gram_det(),
2))
else:
eta_tilde = ZZ(1)
res = {"xi": xi,
"xi_dash": xi_dash,
"xi_hat": xi_hat,
"xi_hat_dash": xi_hat_dash,
"eta_tilde": eta_tilde}
res.update(_invariants_2_common(b1, q2))
return res
def hasse_invariant__OMeara(self):
p = self.p
if p != 2:
rational_diags = [p ** a * b for a, b in self.blocks]
else:
rational_diags = []
for a, b in self.blocks:
if isinstance(b, str):
rational_diags.extend(
[p ** a * x for x in _hy_rational_diags[b]])
else:
rational_diags.append(p ** a * b)
n = len(rational_diags)
res = ZZ(1)
for i in range(n):
for j in range(i + 1, n):
res *= hilbert_symbol(rational_diags[i], rational_diags[j], p)
return res * hilbert_symbol(self.Gram_det(), -1, p)
