Do exist some computational approach to calculation of Tamagawa number for the jacobian of hyperelliptic curve at prime $p$?

As followed from this question one can compute $\Phi(\overline{\mathbb F}_p)$ using genus2reduction Sage function and in case it is trivial one can assert that $\Phi({\mathbb F}_p)$ is trivial too, so $c_p = 1$. If I understand things right same situation with Magma RegularModel approach (see question above and another question), except note that Magma's approach works with $p=2$, but Sage's doesn't.

So question is how to calculate tamagawa number for jacobian of hyperelliptic curve in general case (or at least in some cases) using some computer algebra system like Sage, Magma and etc?


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