Past exam of the geometry course of the University of Oxford / 2011 / 2 / b / ii / Solution

The Riemann-Hurwitz formula gives total ramification four because $0=2\cdot 2-R$. The double pole at $[0]$ has local degree two and contributes one. Since ${\wp }^{\prime }$ is odd and periodic, it vanishes at each nonzero point satisfying $[h]=[-h]$. These areThey are three distinct finite points, each contributing at least one to ramification. Together with $[0]$ they exhaust the total of four. Each therefore has local degree exactly two, meaning its derivative has a simple zero. There are no other zeros of ${\wp }^{\prime }$, and their translates by $\Lambda$ give the full list in the complex plane.