## Abstract

Let D be an integer. Consider the elliptic curve E/ℚ : y^{2} = x^{3} + D, which has j-invariant 0. We can show that for this elliptic curve the rank of its 3-Selmer group is closely related to the 3-rank of the ideal class groups of the quadratic fields ℚ(√D) and ℚ(√-3D). For the same family of curves Frey showed that, if D is a cube, the rank of the Selmer group of a 3-isogeny is related to the class number of the quadratic field ℚ(√D) [3]. Also Jan Nekevář proved some analogous result for elliptic curve given by Dy^{2} = 4x^{3} - 27 which is isomorphic to the curve given by y^{2} = x^{3} - 432 D^{3} [4]. Our method is different from theirs and it can give a far more complete result for general D.

Original language | English |
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Pages (from-to) | 2157-2167 |

Number of pages | 11 |

Journal | Communications in Algebra |

Volume | 25 |

Issue number | 7 |

DOIs | |

State | Published - 1997 |