In the output of genus2reduction, the group of components
$\Phi$ of the Neron model is the group over the algebraic
closure of F_p. The set of rational points of $\Phi$
can be computed using Theorem 1.17 in
S. Bosch and Q. Liu "Rational points of the group of
components of a N\'eron model", Manuscripta Math. 98
(1999), 275-293.
Be carefull that the formula
valuation of the naive minimal discriminant
= f + n - 1 + 11c(X)
(Q. Liu : "Conducteur et discriminant minimal de courbes de
genre 2", Compositio Math. 94 (1994) 51-79, Theoreme 2)
is valid only if the residual field is algebraically
closed as stated in the paper. So this equality does not
hold in general over Q_p. The fact is that the minimal
discriminant may change after unramified extension.
One can show however that, at worst, the change will
stabilize after a quadratic unramified extension
(Q. Liu : "Modeles entiers de courbes hyperelliptiques sur un
corps de valuation discrete", Trans. AMS 348 (1996), 4577-4610,
\S 7.2, Proposition 4).