Picard Curves with Small Conductor

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 one 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 f3 ≥ 4. As an application we discuss the question of finding Picard curves with small conductor.