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)  is in the optimal form, which was the missing case in Sutherland's work (Sutherland, 2012 ).
- Dihedral quartic number field
- Elliptic curve
- Modular curve