Let E and E be elliptic curvesMathworldPlanetmath over a field k. An isogeny between E and E is a finite morphism f:EE of varietiesMathworldPlanetmathPlanetmathPlanetmath that preserves basepoints.

The two curves are called isogenous if there is an isogeny between them. This is an equivalence relationMathworldPlanetmath, symmetryPlanetmathPlanetmath being due to the existence of the dual isogeny. Every isogeny is an algebraicMathworldPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and thus induces homomorphisms of the groups of the elliptic curves for k-valued points.

Title isogeny
Canonical name Isogeny
Date of creation 2013-03-22 12:52:07
Last modified on 2013-03-22 12:52:07
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 14H52
Classification msc 14A15
Classification msc 14A10
Classification msc 14-00
Synonym isogenous
Related topic EllipticCurve
Related topic ArithmeticOfEllipticCurves