# Invariants, Bitangents and Matrix Representations of Plane Quartics with 3-Cyclic Automorphisms

## TL;DR

This paper investigates plane quartic curves with cyclic automorphisms, computing their invariants, bitangents, and matrix representations, revealing specific geometric properties and providing algebraic solutions for their matrix representations.

## Contribution

It computes invariants and bitangents for quartics with cyclic automorphisms and offers a symbolic algebraic solution for their matrix representations.

## Key findings

- Quartics with 6-cyclic automorphism have 3 horizontal bitangents forming an asysgetic triple.
- A degree 6 equation in one variable solves the matrix representation problem for these curves.
- The study links automorphism groups to geometric and algebraic properties of quartic curves.

## Abstract

In this work we compute the Dixmier invariants and bitangents of the plane quartics with 3,6 or 9-cyclic automorphisms, we find that a quartic curve with 6-cyclic automorphism will have 3 horizontal bitangents which form an asysgetic triple. We also discuss the linear matrix representation problem of such curves, and find a degree 6 equation of 1 variable which solves the symbolic solution of the linear matrix representation problem for the curve with 6-cyclic automorphism.