I am using SageMath 10.4.
m = matrix(GF(11), [[1,5,0,0], [5,1,9,0], [0,9,1,5], [0,0,5,1]])
qf = QuadraticForm(m)
Q, T = qf.rational_diagonal_form(return_matrix=True)
T == identity_matrix(GF(11), 4) # True
The transformation T is computed as the identity matrix, which is clearly
wrong.
I think the problem is in
QuadraticForm._rational_diagonal_form_and_transformation, because the
matrix m is singular, so it triggers this section:
except ZeroDivisionError:
# Singular case is not fully supported by PARI
pass
and proceeds to the general case. However, Q has already been modified.
Indeed, we can skip the PARI special case altogether by setting:
setattr(qf, "__pari__", lambda: 1/0) # raises an exception
before computing the rational diagonal form. This gives the correct T.
Thanks.
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