genus of quadratic forms

from sage.quadratic_forms.genera.genus import Genus
from sage.quadratic_forms.genera.genus import GenusSymbol_global_ring
A = DiagonalQuadraticForm(ZZ, [-1, 2, -3, 4, -5, 6, -7, 8, -9]).Hessian_matrix()
Genus(A)

Genus of
[ -2   0   0   0   0   0   0   0   0]
[  0   4   0   0   0   0   0   0   0]
[  0   0  -6   0   0   0   0   0   0]
[  0   0   0   8   0   0   0   0   0]
[  0   0   0   0 -10   0   0   0   0]
[  0   0   0   0   0  12   0   0   0]
[  0   0   0   0   0   0 -14   0   0]
[  0   0   0   0   0   0   0  16   0]
[  0   0   0   0   0   0   0   0 -18]
Signature:  (4, 5)
Genus symbol at 2:    [2^-5 4^2 8^1 16^1]_1
Genus symbol at 3:     1^6 3^2 9^1
Genus symbol at 5:     1^8 5^-1
Genus symbol at 7:     1^-8 7^-1

from sage.quadratic_forms.genera.genus import p_adic_symbol
from sage.quadratic_forms.genera.genus import Genus_Symbol_p_adic_ring

p=3
G3 = Genus_Symbol_p_adic_ring(p, p_adic_symbol(A, p, 2)); G3

Genus symbol at 3:     1^6 3^2 9^1

p=5
G5 = Genus_Symbol_p_adic_ring(5, p_adic_symbol(A, 5, 2)); G5

Genus symbol at 5:     1^8 5^-1

p=7
G7 = Genus_Symbol_p_adic_ring(7, p_adic_symbol(A, 7, 2)); G7

Genus symbol at 7:     1^-8 7^-1