j-invariant of a supersingular elliptic curve

In characteristic $p$, every map $E_1 to E_2$ factors as a power of the Frobenius $varphi_r colon E_1 to E_1^{(p^r)}$ followed by a separable morphism $E_1^{(p^r)} to E_2$, and we find $r$ by looking at the inseparable degree of our map (if the map is separable, then $r=0$, as Pete pointed out).

Now, in the case of interest, if $E$ is supersingular, $widehat{varphi}$ is inseparable (as this is equivalent to multiplication by $p$ being purely inseparable). But then $widehat{varphi} colon E^{(p)} to E$ factors as $E^{(p)} to E^{(p^2)} to E$ by comparing degrees, where the first map is the Frobenius and the second is an isomorphism.

It then follows that $j(E) = j(E^{(p^2)}) = j(E)^{p^2}$ so $j(E) in mathbb{F}_{p^2}$.