## Abstract

We construct infinite families of elliptic curves with cyclic torsion groups over quartic number fields K such that the Galois closure of K is dihedral of degree 8; such a quartic number field K is called a dihedral quartic number field. In fact, all the cyclic torsion groups of elliptic curves which occur over quartic number fields (but not over quadratic number fields) are Z/NZ with N=. 17, 20, 21, 22, 24. The cases of N=. 20 and 24 are treated in the previous work of the authors, and the current work completes the construction of infinite families of elliptic curves over dihedral quartic number fields with cyclic torsion groups (which do not occur over quadratic number fields).

Original language | English |
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Pages (from-to) | 342-363 |

Number of pages | 22 |

Journal | Journal of Number Theory |

Volume | 147 |

DOIs | |

State | Published - 1 Feb 2015 |

### Bibliographical note

Funding Information:The first author was supported by the research grant of the Kongju National University in 2013, the second author by NRF grant funded by the Korea government (MSIP) (No. 2014001824 ), and the third author by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 ) and by the NRF grant funded by the Korea government (MEST) ( 2014-002731 ).

Publisher Copyright:

© 2014 Elsevier Inc.

## Keywords

- Dihedral quartic number field
- Elliptic curve
- Modular curve
- Primary
- Secondary
- Torsion