# examples of elliptic curves with complex multiplication

Here we show some elliptic curves^{} defined over $\mathbb{Q}$ which have complex multiplication^{} by a quadratic imaginary field $K$ of class number^{} $1$ (with $\mathrm{End}(E)$ exactly isomorphic to the full ring of integers^{} ${\mathcal{O}}_{K}$).

$K$ | Curve |
---|---|

$\mathbb{Q}(\sqrt{-1})$ | ${y}^{2}={x}^{3}+x$ |

$\mathbb{Q}(\sqrt{-2})$ | ${y}^{2}={x}^{3}+4{x}^{2}+2x$ |

$\mathbb{Q}(\sqrt{-3})$ | ${y}^{2}+y={x}^{3}$ |

$\mathbb{Q}(\sqrt{-7})$ | ${y}^{2}+xy={x}^{3}-{x}^{2}-2x-1$ |

$\mathbb{Q}(\sqrt{-11})$ | ${y}^{2}+y={x}^{3}-{x}^{2}-7x+10$ |

$\mathbb{Q}(\sqrt{-19})$ | ${y}^{2}+y={x}^{3}-38x+90$ |

$\mathbb{Q}(\sqrt{-43})$ | ${y}^{2}+y={x}^{3}-860x+9707$ |

$\mathbb{Q}(\sqrt{-67})$ | ${y}^{2}+y={x}^{3}-7370x+243528$ |

$\mathbb{Q}(\sqrt{-163})$ | ${y}^{2}+y={x}^{3}-2174420x+1234136692$ |

Title | examples of elliptic curves with complex multiplication |
---|---|

Canonical name | ExamplesOfEllipticCurvesWithComplexMultiplication |

Date of creation | 2013-03-22 14:22:47 |

Last modified on | 2013-03-22 14:22:47 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Example |

Classification | msc 11G05 |

Related topic | ArithmeticOfEllipticCurves |