## Abstract

In this work we establish an effective lower bound for the class number of the family of real quadratic fields Q(d), where d=n ^{2}+4 is a square-free positive integer with n=m(m ^{2}-306) for some odd m, with the extra condition (dN)=-1 for N=2 ^{3}{dot operator}3 ^{3}{dot operator}103{dot operator}10303. This result can be regarded as a corollary of a theorem of Goldfeld and some calculations involving elliptic curves and local heights. The lower bound tending to infinity for a subfamily of the real quadratic fields with discriminant d=n ^{2}+4 could be interesting having in mind that even the class number two problem for these discriminants is not yet solved unconditionally.

Original language | English |
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Pages (from-to) | 2736-2747 |

Number of pages | 12 |

Journal | Journal of Number Theory |

Volume | 132 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Dec 2012 |

Externally published | Yes |

## Keywords

- Class number
- Elliptic curves
- Real quadratic fields

## ASJC Scopus subject areas

- Algebra and Number Theory