Endomorphism ring of supersingular elliptic curve

Question

Let $E$ be a supersingular elliptic curve over $F_q$ where $q=p^n$, then $\operatorname{End}(E)$ is an order in quaterion algebra, hence a non-commutative ring.

Question: Is there an endomorphism $\psi$ of $E$ such that $\psi$ doesn't commute with Frobenius morphism $\phi_q: x\rightarrow x^q$?

Thanks in advance.

Answer 1

Iff $\phi_q\not \in \Bbb{Z}$

(The largest subfields of $End(E)\otimes_\Bbb{Z}\Bbb{Q}$ are $2$-dimensional $\Bbb{Q}$-vector spaces and it is a division ring so any element not in $\Bbb{Q}[\phi_q]$ doesn't commute with $\phi_q$)

Try with $E/\Bbb{F}_9:y^2=x^3+x$ then $\phi_9=-3$ (and $\phi_3\approx \sqrt{-3}$ doesn't commute with $i(x,y)=(-x,iy)$)