I think that kisogenous elliptic curves have the same rank as I think rank is an isogeny invariant. However, I am not sure. Does anyone know where could I find a proof? Thanks!
An isogeny $A \to B$ is a map $A \to B$ with finite kernel. Choose a splitting of $MW(A)$ into torsionfree and torsion summands. This kernel cannot include any of the torsionfree part of $MW(A)$ and so is injective on the torsionfree part so the rank of $MW(B)$ is at least the rank of $MW(A)$. Since the isogeny goes both ways, this gives you equality. 

