# isogeny

Let $E$ and $E^{\prime}$ be elliptic curves over a field $k$. An isogeny between $E$ and $E^{\prime}$ is a finite morphism $f:E\to E^{\prime}$ of varieties that preserves basepoints.

The two curves are called isogenous if there is an isogeny between them. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. Every isogeny is an algebraic homomorphism 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