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, 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 Complex Multiplcation with 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.

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2019-06-13 10:58:13

**Referred to by:**

- ag.modcurve.x0
- dq.ec.reliability
- dq.ecnf.extent
- dq.ecnf.source
- ec.endomorphism
- ec.isogeny_class
- ec.isomorphism
- ec.q.65.a1.bottom
- ec.q.cremona_label
- ec.q.manin_constant
- ec.q_curve
- ec.rank
- lmfdb/ecnf/main.py (line 791)
- lmfdb/elliptic_curves/elliptic_curve.py (line 664)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 573)

**History:**(expand/hide all)

- 2019-06-13 10:58:13 by John Cremona (Reviewed)
- 2018-06-18 21:23:04 by John Jones (Reviewed)

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