Infinite families of elliptic curves over Dihedral quartic number fields

Daeyeol Jeon, Chang Heon Kim, Yoonjin Lee

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Abstract

We find infinite families of elliptic curves over quartic number fields with torsion group Z/NZ with N = 20, 24. We prove that for each elliptic curve Et in the constructed families, the Galois group Gal(L/Q) is isomorphic to the Dihedral group D4 of order 8 for the Galois closure L of K over Q, where K is the defining field of (Et, Qt) and Qt is a point of Et of order N. We also notice that the plane model for the modular curve X1(24) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherland's work (Sutherland, 2012 [12]).

Original languageEnglish
Pages (from-to)115-122
Number of pages8
JournalJournal of Number Theory
Volume133
Issue number1
DOIs
StatePublished - Jan 2013

Bibliographical note

Funding Information:
✩ The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0026917), the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0004284), and the third author was supported by Priority Research Centers Program through the NRF funded by the Ministry of Education, Science and Technology (2010-0028298) and by the NRF grant funded by the Korea government (MEST) (2011-0015684). * Corresponding author. E-mail addresses: dyjeon@kongju.ac.kr (D. Jeon), chhkim@hanyang.ac.kr (C.H. Kim), yoonjinl@ewha.ac.kr (Y. Lee).

Keywords

  • Dihedral quartic number field
  • Elliptic curve
  • Modular curve
  • Torsion

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