Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ and let $C$ be a hyperelliptic curve of genus $g$ defined over $K$ with Jacobian $J$.

Suppose that $C$ is given by an equation $y^2 + h(x)y = f(x)$, where $h$ and $f$ have coefficients in $R$.

The Tamagawa number of $J$ is defined as the order of the group of $k$-rational points of the component group $\Phi$ of a Néron model of $J$ over $R$.

Is it possible to give an upper bound on the Tamagawa number of $J$ in terms of the coefficients of $h$ and $f$?

I'm happy to assume that the equation is minimal in the sense of Liu's paper "Modèles entiers de courbes hyperelliptiques sur un corps de valuation discrète", Trans. of AMS , 348 (1996), 4577-4610 (i.e., the valuation of the discriminant of the given equation of $C$ is minimal among all such equations) and that $k$ is separably closed.