# isogenous elliptic curves have same rank

This is based on exercise 14.3 from Cassels, Lectures on Elliptic Curves. Let $$E:y^2=x(x^2+ax+b), E':y^2=x(x^2+a_1x+b_1)$$ be two elliptic curves over $$\mathbb{Q}$$, with $$a_1=-2a$$, $$b_1=a^2-4b$$. We know that this means there exists a 2-isogeny from $$E$$ to $$E'$$.

Part (a) of the exercise is proving that the groups $$E(\mathbb{Q})$$ and $$E'(\mathbb{Q})$$ have isomorphic odd-order torsion. I managed to do this, however the second part is: Assuming the Mordell-Weil theorem, show that $$E(\mathbb{Q})$$ and $$E'(\mathbb{Q})$$ have the same rank.

I know that $$E(\mathbb{Q}) = E^{\text{tors}}(\mathbb{Q}) \oplus \mathbb{Z}^r$$, where $$r$$ is the rank. How can I use this to prove that the curves have the same rank?

• Use the induced group homomorphism $E(\mathbb Q)\to E'(\mathbb Q)$, which has finite kernel by definition. – Mathmo123 Jun 12 at 8:05
• See this MO-question for an answer. – Dietrich Burde Jun 12 at 9:00