I think that k-isogenous 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!



Rank of isogenous elliptic curves
An isogeny $A \to B$ is a map $A \to B$ with finite kernel. Choose a splitting of $MW(A)$ into torsion-free and torsion summands. This kernel cannot include any of the torsion-free part of $MW(A)$ and so is injective on the torsion-free part so the rank of $MW(B)$ is at least the rank of $MW(A)$. Since whenever there is an isogeny $ f\colon A \to B$ (say of degree $n$), there is an isogeny $B \to A$ (for example, the one whose composition with $f$ is multiplication by $n$) , this gives you equality.


