# Picard curves with small conductor

## TL;DR

This paper investigates the conductor of Picard curves over the rationals, analyzing their bad reduction behavior and establishing lower bounds on the conductor exponent at prime 3, with implications for finding curves with minimal conductor.

## Contribution

It provides a detailed analysis of the stable reduction of Picard curves, establishing restrictions on their conductor exponents and demonstrating that all such curves have bad reduction at 3 with a minimum exponent of 4.

## Key findings

- Picard curves over $Q$ always have bad reduction at p=3.
- The conductor exponent at p=3 is at least 4.
- Restrictions on the conductor exponents help identify curves with small conductors.

## Abstract

We study the conductor of Picard curves over $\mathbb{Q}$, which is a product of local factors. Our results are based on previous results on stable reduction of superelliptic curves that allow to compute the conductor exponent $f_p$ at the primes $p$ of bad reduction. A careful analysis of the possibilities of the stable reduction at $p$ yields restrictions on the conductor exponent $f_p$. We prove that Picard curves over $\mathbb{Q}$ always have bad reduction at $p=3$, with $f_3\geq 4$. As an application we discuss the question of finding Picard curves with small conductor.