$E(\mathbb{F}_q ) \cong \mathbb{Z}/n \mathbb{Z}×\mathbb{Z}/n \mathbb{Z}$ implies $n|q-1$

Question

Let $E$ be an elliptic curve over $\mathbb{F}_q$ with $E(\mathbb{F}_q ) \cong \mathbb{Z}/n \mathbb{Z}×\mathbb{Z}/n \mathbb{Z}$ for some $n \in \mathbb{N}$

I want to show that $n$ divides $q − 1$ and was given the following hint:

What is the image of a basis of $E[n]$ under the Weil pairing?

I think, the image of a basis $\{T_1, T_2\}$ of $E[n]$ under the Weil pairing is a primitive $n$-th root of unity, so if i can show that $e_n(T_1,T_2)^{q-1}=1$, this would imply that $n|q-1$.

However I fail trying to show that. How can I do this?

Answer 1

$E(\mathbb{F}_q ) \cong \mathbb{Z}/n \mathbb{Z}×\mathbb{Z}/n \mathbb{Z}$. Thus $E[n] \subseteq E(\mathbb{F}_q )$, which implies the $n$-th roots of unity $\mu_n \subseteq \mathbb{F}_q $. Now for all $\zeta_n \in \mu_n$ we have $\zeta_n^{q-1} = 1$, thus $n|q-1$.