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.

References in corpus (2)

• Superspecial curves of genus $4$ in small characteristic
• Automorphism groups of superspecial curves of genus $4$ over ${\mathbb{F}}_{11}$

Cited by in corpus (5)

• Computing representation matrices for the action of Frobenius to cohomology groups
• Automorphism groups of superspecial curves of genus $4$ over ${\mathbb{F}}_{11}$
• Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points
• Superspecial trigonal curves of genus $5$
• Counting isomorphism classes of superspecial curves