Enumerating superspecial curves of genus $4$ over prime fields

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 $p \ge 5$\mathrm{.}Weexecutethealgorithmforprimefieldsof$p\le 11$ 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.