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?

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

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