Need this for probabilistic factoring algorithm.

Let $p$ be sufficiently large prime and $E$ the elliptic curve $E /\mathbb{F}_p: y^2=x^3+ax+b$. Let $o=\#E(\mathbb{F}_p)$. $\psi_n$ denote the $n$-the division polynomial of $E$.

If $X$ is the $x$ coordinate on $E$ we have $\psi_o(X)=0$.

What hypothesis on $E$ do we need to make it only if?

$$\psi_o(X)=0 \iff X^3+aX+b=\square$$

I suspect it is enough $E$ to not be supersingular.

Would prefer reference rather than proof.