Abstract

In this paper, we enumerate superspecial trigonal curves of genus 5 over ${\mathbb{F}}_{p^a}$ for any natural number a if $p\le 7$ and for odd a if $p\le 13.$ This paper provides an algorithm enumerating superspecial trigonal curves of genus 5 over ${\mathbb{F}}_q$ for arbitrary q. We implemented the algorithm over computer algebra system Magma. With help of computational results obtained by our implementation, we prove the nonexistence (resp. existence) of superspecial trigonal curves of genus 5 over ${\mathbb{F}}_q$ for q = 7, 7² and 13 (resp. q = 11). Executing the algorithm also determines the number of isomorphism classes of such curves for q = 11 with explicit defining equations.

摘要

在本文中，我们列举了第 5 类的超特殊三角曲线${\mathbb{F}}_{p^{\mathrm{一个}}}$对于任何自然数a if$p\le 7$和奇怪的一个如果$p\le 13.$本文提供了一种枚举第 5 类超特殊三角曲线的算法${\mathbb{F}}_q$对于任意q。我们在计算机代数系统 Magma 上实现了该算法。借助我们的实现获得的计算结果，我们证明了第 5 类超特殊三角曲线的不存在（或存在）${\mathbb{F}}_q$对于q  = 7、7 ²和 13（分别为q  = 11）。 执行该算法还使用明确的定义方程确定q = 11的此类曲线的同构类的数量。