Degree-two even elliptic function ($f=(a\wp +b)/(c\wp +d)$)

An even elliptic function descends through the degree-two quotient map $\wp$ to a rational function $R$ on the Riemann sphere. If the original function also has degree two, composition of map degrees gives $2=2\mathrm{deg}R$, so $R$ has degree one. Consequently it is a Möbius transformation with $ad-bc\ne 0$. The shift $\mathrm{sn}(z+K,k)$ is even and has degree two, providing an example for the common period lattice of the Jacobi elliptic sine and $\wp$.