Let $W \subseteq \mathbb{A}^{2}$ be the curve defined by the equation $y^{3}=x^{4}+1$ over the complex numbers $\mathbb{C}$, and let $X \subseteq \mathbb{P}^{2}$ be its closure.

(a) Show $X$ is smooth.

(b) Determine the ramification points of the $\operatorname{map} X \rightarrow \mathbb{P}^{1}$ defined by

$$(x: y: z) \mapsto(x: z) .$$

Using this, determine the Euler characteristic and genus of $X$, stating clearly any theorems that you are using.

(c) Let $\omega=\frac{d x}{y^{2}} \in \mathcal{K}_{X}$. Compute $\nu_{p}(\omega)$ for all $p \in X$, and determine a basis for $\mathcal{L}\left(\mathcal{K}_{X}\right)$