Exceptional elliptic curves over quartic fields

We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = Z/mZ ⊕ Z/nZ, where m|n, be a torsion group such that the modular curve X1(m,n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as… (More)