Enumerating superspecial curves of genus $4$ over prime fields

Abstract

In this paper we enumerate nonhyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $p\le 11$. Our algorithm works for nonhyperelliptic curves over an arbitrary finite field in characteristic $p \ge 5$. We execute the algorithm for prime fields of $p\le 11$ with our implementation on a computer algebra system Magma. Thanks to the fact that the cardinality of $\mathbb{F}_{p^a}$-isomorphism classes of superspecial curves over $\mathbb{F}_{p^a}$ of a fixed genus depends only on the parity of $a$, this paper contributes to the odd-degree case for genus $4$, whereas our previous paper contributes to the even-degree case.