Branching data for curves up to genus 48

Abstract

An algorithm of Thomas Breuer produces complete lists of automorphism groups of curves for a fixed genus, and the code to execute this algorithm was written in the computer algebra package GAP and run by Breuer over a decade ago. For each curve $X$ of genus $g$ and a group $G$ acting on $X$, the branching data for the covering $X \to X/G$ is computed but was not record. We have added functionality to the code to output this data, and have translated the code to the computer algebra package Magma. This article provides an explanation of the code to add this branching data. Complete data from this modified program for curves up to genus 20, and data for genus up to 48 of groups $G$ so that $|G|>4(g-1)$ are available. We also include sample programs which provide a way to search the data for groups or actions with desired properties, these functions should be of use to researchers who want to find examples of group actions on lower genus curves with certain properties, or who need explicit examples of generating vectors for their research. The code also may be used to create generating vectors for higher genus curves, if the automorphism group and signature of the mapping $X \to X/G$ are already know.