Elliptic curves with all cubic twists of the same root number

Let E/KE/K be an elliptic curve with j-invariant 0 defined over a number field K. In this paper, we give a simple condition on K   which determines whether all cubic twists of E/KE/K have the same root number or not. This is a cubic twist analogue to the work [D-D] of Dokchitser and Dokchitser on quadratic twists of elliptic curves.