= Homomorphism group of elliptic curves
{title2=$\operatorname{Hom}_K(E_1,E_2)$}
The group $\operatorname{Hom}_K(E_1,E_2)$ consists of morphisms of <elliptic curves> defined over $K$ that preserve the <group operation>, with pointwise addition. Its zero element is the constant map to the identity. Every nonzero element is an <isogeny of elliptic curves>: its image is a closed connected subgroup of the target, so nonconstancy makes it the whole curve, and a nonconstant morphism of smooth projective curves is finite. The <degree of an isogeny> extends to this group by assigning degree zero to the zero homomorphism.
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