Determining equations of families of cyclic curves

TL;DR

This paper derives explicit parametric equations for families of cyclic algebraic curves with specified automorphism groups over fields of odd characteristic, expanding understanding of their algebraic structure.

Contribution

It provides the first comprehensive set of parametric equations for these families, linking automorphism groups to explicit curve equations.

Findings

Explicit parametric equations for each automorphism group family

Classification of cyclic curves based on automorphism groups

Extension of previous automorphism group results

Abstract

In previous work we determined automorphism groups of cyclic algebraic curves defined over fields of any odd characteristic. In this paper we determine parametric equations of families of curves for each automorphism group for such curves.