Isogeny of elliptic curves

An isogeny is a nonconstant morphism of elliptic curves preserving their identity points. It is a finite surjective group homomorphism.

For an isogeny $\varphi :E\to E^{\prime }$, its kernel is denoted $E[\varphi ]$. In characteristic zero its cardinality equals $\mathrm{deg}\varphi$.

The dual of an isogeny $\varphi :E\to E^{\prime }$ of degree $n$ is the unique isogeny $\hat{\varphi }:E^{\prime }\to E$ satisfying

For $\varphi \in \mathrm{End}(E)$, its trace is the integer characterized byEquivalently, $\mathrm{tr}(\varphi )=1+\mathrm{deg}\varphi -\mathrm{deg}(1-\varphi )$.

For an elliptic curve over ${\mathbb{F}}_q$, the Frobenius isogeny sends $(x,y)$ to $(x^q,y^q)$. If $a=q+1-\#E({\mathbb{F}}_q)$, then

If $\alpha ,\beta$ are the roots of $T^2-aT+q$, where $a=q+1-\#E({\mathbb{F}}_q)$, then

For an elliptic curve over ${\mathbb{F}}_q$,

If $a=q+1-\#E({\mathbb{F}}_q)$, then