Even elliptic functions are rational in the Weierstrass function ($f(-z)=f(z)⟹f=Q(\wp )$)

The Weierstrass elliptic function is the quotient map from the complex torus by $z\mapsto -z$. An even elliptic function descends through this map; its even Laurent series at each branch point makes the descended function meromorphic there. A meromorphic function on the Riemann sphere is rational, so $f=Q(\wp )$.

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 $\wp$.