# 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 |