Hi, overflowers.

I have a question concerning the torsion of elliptic curves over number fields.

Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can prove that, for all primes $p$, ${\mathbb Q}[\zeta_p] \subset {\mathbb Q}(E[p])$, where ${\mathbb Q}(E[p])$ is the algebraic extension generated by the coordinates of all points of order $p$ on $E$.

I was thinking whether this could also be true for all integers $n$, not necessarily prime. It is probably something well-known, but I can not find an appropriate reference, nor an easy counterexample.

Thanks a lot!!