Let $X$ be a smooth projective curve over an algebraically closed field $k$.

State the Riemann-Roch theorem, briefly defining all the terms that appear.

Now suppose $X$ has genus 1 , and let $P_{\infty} \in X$.

Compute $\mathcal{L}\left(n P_{\infty}\right)$ for $n \leqslant 6$. Show that $\phi_{3 P_{\infty}}$ defines an isomorphism of $X$ with a smooth plane curve in $\mathbb{P}^{2}$ which is defined by a polynomial of degree 3 .