Isogeny

The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind or nature".

is called degree of isogeny: Properties of degree: For abelian varieties, such as elliptic curves, this notion can also be formulated as follows: Let E1 and E2 be abelian varieties of the same dimension over a field k. An isogeny between E1 and E2 is a dense morphism f : E1 → E2 of varieties that preserves basepoints (i.e. f maps the identity point on E1 to that on E2).

This is equivalent to the above notion, as every dense morphism between two abelian varieties of the same dimension is automatically surjective with finite fibres, and if it preserves identities then it is a homomorphism of groups.

This can be shown to be an equivalence relation; in the case of elliptic curves, symmetry is due to the existence of the dual isogeny.

As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.