I am trying to compute the group structure as the title says of $E:y^2=x^3 - x$ over $\mathbb{F}_p$ with $p\equiv 1\bmod 8$ and $p-1$ a square, that is, $p=(a+i)(a-i)$ over $\mathbb{Z}[i]$.
I just proved that $\#E(\mathbb{F}_p)=p + 1 - 2 = p - 1 = a^2$ and in MAGMA it says that as an abstract group for some examples is isomorphic to $\mathbb{Z}/(a)\times \mathbb{Z}/(a)$. I spent some time but I cannot get more far than the cardinality yet (I thought it was going to be simpler or there is something I am not seeing).
This curve in this case has CM by $i$ that is, $\text{End}_{\mathbb{F}_p}(E)=\mathbb{Z}[i]$. Another trivial observation is that it has full $2$-torsion, hence the $E(\mathbb{F}_p)$ cannot be cyclic.
Any hints will be appreciated, maybe I have to use the Weil Pairing.