What is polynomial $f(X)\in \mathbb{F}_{p^2}$ s.t. $f(j)=0$ iff $j$ is supersingular.

Question

Sorry for my bad English.

Let $p$ be a prime.

We say $j\in \mathbb{F}_{p^2}$ is supersingular if the corresponding elliptic curve to $j$ is supersingular.

So there is polynomial $f(X)\in \mathbb{F}_{p^2}[X]$ s.t. $f(j)=0$ iff $j$ is supersingular.

I have two questions.

First, $f(X)\in \mathbb{F}_p$?

Second, does $f(X)$ have another equivalent definition?

Answer 1

Yes, here is a paper addressing exactly this question:

Luís R. A. Finotti. "A formula for the supersingular polynomial." Acta Arithmetica 139.3 (2009): 265-273. http://eudml.org/doc/278683.