We construct infinite families of elliptic curves with cyclic torsion groups over quartic number fields K such that the Galois closure of K is dihedral of degree 8; such a quartic number field K is called a dihedral quartic number field. In fact, all the cyclic torsion groups of elliptic curves which occur over quartic number fields (but not over quadratic number fields) are Z/NZ with N=. 17, 20, 21, 22, 24. The cases of N=. 20 and 24 are treated in the previous work of the authors, and the current work completes the construction of infinite families of elliptic curves over dihedral quartic number fields with cyclic torsion groups (which do not occur over quadratic number fields).
- Dihedral quartic number field
- Elliptic curve
- Modular curve