  In a series of papers we classify the possible torsion structures of rational
elliptic curves base-extended to number fields of a fixed degree. In this paper
we turn our attention to the question of how the torsion of an elliptic curve
with complex multiplication defined over the rationals grows over quadratic
fields. We go further and we give an explicit characterization of the quadratic
fields where the torsion grows in terms of some invariants attached to the
curve.
