BINARY QUARTIC FORMS WITH BOUNDED INVARIANTS AND SMALL GALOIS GROUPS

We consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms f (x, y) of nonzero discriminant. Then, we shall enumerate the GL(2)(Z)-equivalence classes of all such forms associated to a fixed f (x, y).