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Let $E_1$ and $E_2$ be two elliptic curves defined over the 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., 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.

Isogeny is an equivalence relation, and the equivalence classes are called isogeny classes. Over a number field, isogeny classes are finite.

Isogenies from an elliptic curve $E$ to itself are called endomorphisms.

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

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  • Last edited by John Jones on 2018-06-18 21:23:04
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