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 E_t 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 (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 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]).