Isogeny is morphism between elliptic curves which keeps base point.
Then, is every isogeny between elliptic curves bijective?
Let $E1$ and $E2$ be isogenous elliptic curves defined over Fq. Then #E1(Fq) = #E2(Fq) holds, but the isogeny is not always bijective?
Thank you in advance.
No, isogenies need not be bijective. For example $[2]$ is an isogeny from any elliptic curve $E$ to itself. Its kernel is the 2-torsion $E[2]$. Therefore it is not bijective whenever $E(\Bbb{F}_q)$ has non-trivial 2-torsion.