Let $q$ be a prime power, so that $\mathbb F_q$ is the field on $q$ elements, and suppose $E$ is an elliptic curve with group structure $E(\mathbb F_q) \simeq (\mathbb Z/n\mathbb Z) \times (\mathbb Z/n\mathbb Z)$.
I want to show that either \begin{align*} q&=n^2 + 1, \\ q&=n^2 \pm n +1, \text{ or}\\ q&=n^2\pm 2n + 1 = (n^2\pm 1), \end{align*} but I'm not sure how to go about this.
I'm guessing I need to use some reasoning with torsion points of the curve, but I'm not sure exactly how to proceed. I appreciate any help.