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 / N Z with N = 20 , 24 . We prove that for each elliptic curve E t in the constructed families, the Galois group Gal ( L / Q ) is isomorphic to the Dihedral group D 4 of order 8 for the Galois closure L of K over Q , where K is the defining field of ( E t , Q t ) and Q t is a point of E t of order N . We also notice that the plane model for the modular curve X 1 ( 24 ) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherlandʼs work (Sutherland, 2012 [12] ).