  Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic
curves over $\mathbb{F}_p$ in terms of the level of the defining cubic
homogenous polynomial. We extend their study to arbitrary genus, in particular
we prove that every ordinary hyperelliptic curve $\mathcal{C}$ of genus $g\geq
2$ has level $2$. We provide a good number of examples and raise a conjecture.
