Elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup Show that an elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup.
We've been told that for this problem, we are not allowed to use Mazur's Theorem. Unfortunately that is the only way I can think to answer this question. It was suggested that a geometric argument can be made. Can someone point me in a proper direction? I was thinking of using Nagell-Lutz to try to show there may be more than 3 elements with an order 2. I'm not sure it can be done though.

Hint: the existence of the Weil pairing.
