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 $f=Q(\wp )$.