The group of points of an elliptic curve $E$ over a finite field $E(\mathbb{F}_q)\cong \mathbb{Z}/n\mathbb{Z} \times\mathbb{Z}/n \mathbb{Z}$

Question

For the group of an elliptic curve $E$ over a finite field $\mathbb{F}_q$. If $E(\mathbb{F}_q)\cong \mathbb{Z}/n\mathbb{Z} \times\mathbb{Z}/n \mathbb{Z}$, then $q=n^2+1$ or $q=n^2\pm n +1$ or $q=(n\pm 1)^2$.

I would like to prove the above statement. How could I proceed from $n|p-1$ to deduce the above result?

Answer 1

"Could you give me some hint about this problem?". I have a hint, but it is not complete: Let $E$ be an elliptic curve over $\Bbb F_q$. Then the Weil Pairing $E(\Bbb F_q) \times E(\Bbb F_q) \rightarrow \Bbb F_q^{\ast}$ shows that there exist positive integers $n_1$,$n_2$ such that $$ E(\Bbb F_q) \cong \Bbb Z/n_1 \times \Bbb Z/n_2 $$ with $n_1 \mid {\rm gcd}(n_2,q-1)$. Since we have $n_1=n_2=n$, we have $n\mid q-1$.