Elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup

Question

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.

Answer 1

Hint: the existence of the Weil pairing.

Answer 2

We can use $\mathbf{R}$ in place of $\mathbf{Q}$, if we like.

Draw a picture of a curve that has three real $2$-torsion points in Weierstrass form. How many of the $2$-torsion points can be the double of some other point?

Recall that every line intersects an elliptic curve in exactly three points, counting multiplicity and the point at infinity.