Is every elliptic curve isogeny over $\mathbb{F}_p$ separable?

Question

I know that a isogeny $\varphi: E \rightarrow E^\prime$ has degree equal to its kernel if and only if the isogeny is separable.

I want to know if this always holds for elliptic curves over $\mathbb{F}_p$, with $p$ prime. If this isn't true I would like an example of a non-separable isogeny over $\mathbb{F}_p$.

Answer 1

The Frobenius endomorphism $F:E\to E$ given by $F:(x,y)\mapsto(x^p,y^p)$ when $E$ is in Weierstrass form is inseparable.