Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 23G / ii

Let $P_j=({\lambda }_j,0)$, and let $Q_{\pm }=(0,\pm \sqrt{-{\lambda }_1{\lambda }_2{\lambda }_3})$. The latter are distinct because the product is nonzero and the characteristic is not two. At each $Q_{\pm }$, the derivative of the curve equation with respect to $y$ is nonzero, so $x$ is a local parameter and has a simple zero. At $P_j$, the derivative of the cubic in $x$ is nonzero, so $y$ is a local parameter with a simple zero; $x-{\lambda }_j$ has order two there.

The only poles occur at the point at infinity, and their orders were computed in part (i). Thus the principal divisors areOver a field where the $Q_{\pm }$ are not rational, their sum is interpreted as the degree-two zero divisor, or computed after scalar extension and then descended.