A family of monogenic S4 quartic fields arising from elliptic curves

JOURNAL OF NUMBER THEORY

卷:197

Article

Gassert, T. Alden¹  Smith, Hanson²  Stange, Katherine E.² 

[1] Hobart & William Smith Coll, Dept Math & Comp Sci, 300 Pulteney St, Geneva, NY 14456 USA

[2] Univ Colorado, Dept Math, Campus Box 395, Boulder, CO 80309 USA

关键词: Elliptic curves;    Rings of integers;    Division polynomials;    Monogenicity;    Monogeneity;    Torsion fields;    Division fields;   

DOI  :  10.1016/j.jnt.2018.09.026

来源: Elsevier

【 摘 要 】

We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial 3-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic S-4 fields given by T-4-6T(2)-alpha T-3 for alpha is an element of Z such that alpha +/- 8 are squarefree, are monogenic. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free