(Joint work with Irene Bouw and Michel Boerner.) Let K by a number field, g >= 1 and C>0. It follows from the Shafarevich conjecture for abelian varieties (a theorem of Faltings) that there are at most finitely many smooth projective curves of genus g over K with conductor bounded by C. In my talk I will report about our attempt to make this result more explicit for Picard curves over the rationals.