Infinite families of elliptic curves over Dihedral quartic number fields

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 Sutherlandʼs work (Sutherland, 2012 [12]).