Let $f:X\to \mathbf{P}^1$ be a branched cover of smooth projective connected curves over $\overline{\mathbf{Q}}$, where $\mathbf{P}^1$ is the projective line over $\overline{\mathbf{Q}}$.
What relations are there between the discriminant of the function field extension associated to $f:X\to \mathbf{P}^1$ and other more common "invariants" of the cover like the ramification divisor, etc. ?