Let $E$ be an elliptic curve over algebraically closed field of characteristic $p$. Then, according to Silverman's book ' the arithmetic of elliptic curves', $E$ is supersingular(dual of $p^r$-Frobenius is purely inseparable for all $r≧1$) if only if $[p]$ is purely inseparable and $j(E)∈$$\Bbb F_q $, where $q＝p^2$.
But is $j(E)∈$$\Bbb F_q $$(q＝p^2)$ needed?
In the proof of showing $[p]$ is super singular and $j(E)∈$$\Bbb F_q $$(q＝p^2)$⇨ $E$ is super singular, It seems we do not used the condition $j(E)∈$$\Bbb F_q $$(q＝p^2)$.
(I may have overlooked the process using $j(E)∈$$\Bbb F_q $$(q＝p^2)$)
In the showing process ' $[p]$ is super singular and $j(E)∈$$\Bbb F_q $$(q＝p^2)$⇨ $E$ is super singular' , where did he used the condition $ j(E)∈$$\Bbb F_q $$(q＝p^2)$?