@article{fafbdfc24c6949ee8f97facd48a3c0c3,

title = "Infinite families of elliptic curves over Dihedral quartic number fields",

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]).",

keywords = "Dihedral quartic number field, Elliptic curve, Modular curve, Torsion",

author = "Daeyeol Jeon and Kim, {Chang Heon} and Yoonjin Lee",

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).",

year = "2013",

month = jan,

doi = "10.1016/j.jnt.2012.06.014",

language = "English",

volume = "133",

pages = "115--122",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

number = "1",

}