Let $~E:y^2=x^3+x~$ be an elliptic curve over finite field $\mathbb{F}_{5},$ I compute the trace of Frobenius is $2$($E/\mathbb{F}_{5}$ obvious is ordinary).
(By the theory of CM, I know (when $E$ is an ordinary elliptic curve over finite field $\mathbb{F}_{q}$):$$\mathbb{Z}[\pi_{E}]\subseteq End(E)\subseteq \mathcal{O}_{K}$$ )
in the above example I got $\mathcal{O}_{K}=\mathbb{Z}[i]$, and $\mathbb{Z}[\pi_{E}]=\mathbb{Z}+2\mathcal{O}_{K}.$
Therefore $End(E)$ have two possible result: $\mathcal{O}_{K}$ or $\mathbb{Z}[\pi_{E}]$.
I want to know which is the true $End(E)$.
Actually, there is a bound for the conductor $[\mathcal{O}_{K}:\mathbb{Z}[\pi_{E}]].$ I'am not sure whether or not there exists any better result for deciding the conductor.
In your case, $End(E)=\mathcal O_K$, because $E$ has an automorphism of order $4$, namely $x\mapsto -x$ and $y\mapsto -2y$. This endomorphism cannot belong to $\mathbb Z[\pi_E]$, because you can check that there are no elements of $\mathbb Z[\pi_E]$ whose square is $-1$.