Let $p$ be prime and $q = p^n$. Let $E$ be an elliptic curve over $\mathbb{F}_q$, and let $E^{(p)}$ be the pullback of $E$ by the $p$-power Frobenius of $\mathbb{F}_q$. If $E$ is isomorphic (over $\mathbb{F}_q$) to its Galois conjugate $E^{(p)}$, then does it follow that $E$ is the base change of an elliptic curve over $\mathbb{F}_p$? If so, what is the argument?

Note that similar statements are not true over infinite fields: for instance, $\mathbb{Q}$-curves are isomorphic to all their Galois conjugates, yet need not descend to actual elliptic curves over $\mathbb{Q}$.

isogenousto all their Galois conjugates, not necessarily isomorphic. $\endgroup$