Proof that "Two elliptic curves which are birationally equivalent are isomorphic".

Question

I am looking for the proof of the following statement: Two elliptic curves which are birationally equivalent are isomorphic. Hopefully someone can mention to me some material to prove it. Thanks in advance!

Answer 1

This is a special case of a more general statement: If $C$ and $C'$ are smooth projective curves and $f : C \to C'$ is a rational map, then $f$ is a morphism.

See here, or see Hartshorne I.6.