# Conductor and discriminant of Picard curves

## TL;DR

This paper studies Picard curves, providing normal forms, minimal models, and classifying special Picard curves over rationals with limited bad reduction, including a database of small conductor examples.

## Contribution

It introduces a comprehensive classification of Picard curves over  with specific arithmetic properties and compiles a database of small conductor cases.

## Key findings

- All special Picard curves over  with good reduction outside 2 and 3 identified.
- Smallest conductor for a special Picard curve determined.
- Database of Picard curves with small conductor compiled.

## Abstract

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $\mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have. We also collect a database of Picard curves over $\mathbb{Q}$ of small conductor.