Group Structure for $E(\overline{\mathbb{F}_q})$ Consider an elliptic curve $E$ with coordinates over a field. We know for a fact that for the group operation on the curve the points form an abelian group. The structure of any abelian group can be written as a product of cyclic groups, and the possible groups are classified when the field is finite.
The thing is, what about the points with coordinates over $\overline{\mathbb{F}_q}$? Is it determined by the structure over $\mathbb{F}_q$? It seems like an easy question, but i couldn't find anything about it on the internet.
 A: 
$E(\overline{\mathbb{F}_q})$ is isomorphic to $ \Bbb{Z_{(p)}^2/Z^2}$ or $ \Bbb{(Q\times Z_{(p)})/Z^2}$  depending on if the curve is supersingular or not, where $\Bbb{Z}_{(p)}$ are the rational numbers with denominator not divisible by $p$. 

(the proof is based on that the endomorphism $[n]$ is separable of degree $n^2$ when $p\nmid n$, thus $\#E[n] = n^2$. For the case $p| n$, $\# E[p]$ is the separable degree of the endomorphism $[p]$, since the $p$-th power Frobenius $\phi$ is inseparable and has degree $p$, with $\phi^*$ its dual isogeny then $[p]=\phi^*\phi$ is not separable and since its degree is $p^2$, its separable degree is $p$ or $1$)
If $E[p]=\langle R\rangle$ (non-supersingular) then $Gal(\Bbb{F}_q(E[p])/\Bbb{F}_q)$ is a cyclic subgroup of $GL(E[p])=GL_1(\Bbb{F}_p)$, thus $Gal(\Bbb{F}_q(E[p])\subset \Bbb{F}_{q^{p-1}}$.
Whence, to see if the curve is non-supersingular, it suffices to check if $p$ divides $\#E(\Bbb{F}_{q^{p-1}})=q^{p-1}+1-\alpha^{p-1}-\beta^{p-1}$ where $t=\#E(\Bbb{F}_q)-q-1$ and the minimal polynomial of the Frobenius is $X^2-t X+q=(X-\alpha)(X-\beta)$. 
Note that a complex elliptic curve is isomorphic to $\Bbb{R^2/Z^2}$ and the torsion part is isomorphic to $\Bbb{Q^2/Z^2}$.
