Equations for superelliptic curves over their minimal field of definition

TL;DR

This paper derives explicit equations for superelliptic curves over their minimal fields of definition, especially when the curves have additional automorphisms, using dihedral invariants and automorphism group properties.

Contribution

It provides a method to construct equations of superelliptic curves over their minimal fields of definition leveraging dihedral invariants and automorphism group analysis.

Findings

Explicit equations over minimal fields of definition for superelliptic curves with extra automorphisms

Use of dihedral invariants to facilitate equation construction

Enhanced understanding of automorphism groups in the context of minimal fields

Abstract

Let $\mathcal X_g$ be a genus $g\geq 2$ superelliptic curve, $F$ its field of moduli, and $K$ the minimal field of definition. In this short note we construct an equation of the curve $\mathcal X_g$ over its minimal field of definition $K$ when $\mathcal X_g$ has extra automorphisms. We make use of the dihedral invariants of superelliptic curves as defined by Shaska in [6] and results on the automorphism groups of superelliptic curves as in [10].