q-torsion points of nonsingular elliptic curve

Question

Suppose we have non-supersingular elliptic curve $E$ over $GF(q)$.

How to find minimal k that $|E(GF(q^k)[q]| = q$?

Answer 1

Write $q = p^r$ with $p$ prime. If $r=1$ then $k$ is the multiplicative order mod $p$ of the eigenvalue $\alpha$ of Frobenius that is coprime with $p$, or alternatively the multiplicative order mod $p$ of the Hasse invariant (I assume you meant non-supersingular when you wrote non-singular). For general $r$, first find $k=k_0$ that solves the problem with $r=1$ and $s$ such that $p^s$ is the highest power of $p$ dividing $\alpha^{k_0} -1$. Then the answer is $k=k_0p^{r-s}$, I think.