Let $f \in k[x]$ be a polynomial with distinct roots, $\operatorname{deg} f=d>2$, char $k=0$, and let $C \subseteq \mathbf{P}^{2}$ be the projective closure of the affine curve

$$y^{d-1}=f(x)$$

Show that $C$ is smooth, with a single point at $\infty$.

Pick an appropriate $\omega \in \Omega_{k(C) / k}^{1}$ and compute the valuation $v_{q}(\omega)$ for all $q \in C$.

Hence determine $\operatorname{deg} \mathcal{K}_{C}$.