Wild conductor exponents of curves

We give an explicit formula for wild conductor exponents of plane curves over $\mathbb{Q}_p$ in terms of standard invariants of explicit extensions of $\mathbb{Q}_p$, generalising a formula for hyperelliptic curves. To do so, we prove a general result relating the wild conductor exponent of a simply branched cover of the projective line with its associated discriminant cover. In an appendix, we resolve a minor issue in the literature on the $3$-torsion of genus 2 curves.