### Abstract

We study the ramifications in the extensions of number fields arising from an isogeny of elliptic curves. In particular, we start with an elliptic curve with a rational torsion point, and show that the extension is unramified if and "only if" the point which generates the extension is reduced into a nonsingular point (we need to assume certain conditions in order to prove the "only if" part). We also study a characterization of quadratic number fields with class numbers divisible by 5.

Original language | English |
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Pages (from-to) | 809-826 |

Number of pages | 18 |

Journal | Osaka Journal of Mathematics |

Volume | 48 |

Issue number | 3 |

Publication status | Published - 2011 Sep 1 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Sato, A. (2011). On the class numbers of certain number fields obtained from points on elliptic curves III.

*Osaka Journal of Mathematics*,*48*(3), 809-826.