Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 3 / b / Solution

Write $X=x\circ \varphi$. Since $X(-P)=X(P)$, it is an even rational function and therefore belongs to $k(x)$; write $X=p(x)/q(x)$ with coprime polynomials $p,q$. The pullback of a nonzero invariant differential is invariant, so for some $c\in k^{\times }$,Sincewe obtainThus