Isogeny between elliptic curves (reviewed)

Let $E_1$ and $E_2$ be two elliptic curves defined over a field $K$. An isogeny (over $K$) between $E_1$ and $E_2$ is a non-constant morphism $f\colon E_1 \to E_2$ defined over $K$, i.e., a morphism of curves given by rational functions with coefficients in $K$, such that $f(O_{E_1})= O_{E_2}$. Elliptic curves $E_1$ and $E_2$ are called isogenous if there exists an isogeny $f\colon E_1 \to E_2$.

An isogeny respects the group laws on $E_1$ and $E_2$, and hence determines a group homomorphism $E_1(L)\to E_2(L)$ for any extension $L$ of $K$. The kernel is a finite group, defined over $K$; in general the points in the kernel are not individually defined over $K$ but over a finite Galois extension of $K$ and are permuted by the Galois action.

The degree of an isogeny is its degree as a morphism of algebraic curves. For a separable isogeny this is equal to the cardinality of the kernel. Over a field of characteristic $0$ such as a number field, all isogenies are separable. In finite characteristic $p$, isogenies of degree coprime to $p$ are all separable.

An isogeny is cyclic if its kernel is a cyclic group. Every isogeny is the composition of a cyclic isogeny with the multiplication-by-$m$ map for some $m\ge1$.

Isogeny is an equivalence relation, and the equivalence classes are called isogeny classes. Over a number field, it is a consequence of a theorem of Shafarevich that isogeny classes are finite. Between any two curves in an isogeny class there is a unique degree of cyclic isogeny between them, except when the curves have additional endomorphisms defined over the base field of the curves; in that case there are cyclic isogenies of infinitely many different degrees between any two isogenous curves.

Isogenies from an elliptic curve $E$ to itself are called endomorphisms. The set of all endomrpshisms of $E$ forms a ring under pointwise addition and composition, the endomorphism ring of $E$.

An isogeny of elliptic curves is a special case of an isogeny of abelian varieties.