I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -

1) $E_{1}, E_{2}$ have no rational torsion points.

2) $E_1[9] \cong E_2[9]$ as Gal$(\overline{\mathbb{Q}} / \mathbb{Q})$ modules.

I will request you to kindly suggest how to start finding out examples having the above two properties or suggest some examples. If necessary, I am comfortable with basic Sage computations.


1 Answer 1


In general, for any integer $N$ and any fixed elliptic curve $E$, the elliptic curves $E'$ for which $E[N]\cong E'[N]$ as Galois modules (and such that the isomorphism respects the Weil pairing) are parametrised by a twist $Y_E(N)$ of the modular curve $Y(N)$.

Tom Fisher has worked out equations of these twists for $N=7,9$ and 11. He explicitly writes down examples of such pairs of elliptic curves (and the very first example in section 7.2 has trivial torsion), and shows that there are infinitely many pairs satisfying your condition 2), together with the additional condition that the isomorphism preserves (respectively "anti-preserves") the Weil pairing. Since he gives his families in parametric form, you may be able to restrict the parameter to ensure that the torsion subgroup is trivial.

If you want to go and search for such pairs yourself, you can start with any elliptic curve with trivial torsion, and search for points on $Y_E(N)$, using Tom Fisher's equation and brute force.

Edit: Tom also has a fairly large table of pairs of curves satisfying your condition 2).

Edit II, regarding $N=3$: The (compactifications of) the twists $Y_E(3)$ have genus 0, and moreover, they each have a rational point, given by $E$ itself. So for any fixed $E$, you can find infinitely many $E'$ with isomorphic 3-torsion, all you need to know is the equation of $Y_E(3)$. For that, see the papers of Rubin and Silverberg, referenced on the first page of Tom Fisher's paper.

  • $\begingroup$ Is there any table for $N=3$ i.e. $E_1[3] \cong E_2[3]$ as Gal$(\overline{\mathbb{Q}} / \mathbb{Q})$ modules ? $\endgroup$
    – Suman
    Apr 29, 2014 at 7:12

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