Why is the rank of elliptic curves isogeny invariant?

Question

Let $E/K$ and $E'/K$ be an elliptic curve over a field.

If there exists an be an isogeny of elliptic curve $\phi: E\to E'$, why can we say $rank(E/K)=rank(E'/K)$ ?

Here, I define as non constant morphism which sends base point to base point.

Answer 1

The kernel of the isogeny is a group of points of finite order equal to the degree. Therefore the free part of the group of points on $E$ is the same size after the isogeny: it only affects the torsion part.