Let $E \subseteq \mathbf{P}^{2}$ be the projective curve obtained from the affine curve $y^{2}=\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)\left(x-\lambda_{3}\right)$, where the $\lambda_{i}$ are distinct and $\lambda_{1} \lambda_{2} \lambda_{3} \neq 0$.

(i) Show there is a unique point at infinity, $P_{\infty}$.

(ii) Compute $\operatorname{div}(x), \operatorname{div}(y)$.

(iii) Show $\mathcal{L}\left(P_{\infty}\right)=k$.

(iv) Compute $l\left(n P_{\infty}\right)$ for all $n$.

[You may not use the Riemann-Roch theorem.]