Elliptic curves with all quartic twists of the same root number

Abstract

Let $E/K$ be an elliptic curve with $j$-invariant 1728 defined over a number field $K$. In this note, we give a simple condition on $K$ which determines whether all quartic twists of $E/K$ have the same root number or not. This completes a series of works on the same root number of twists begun in [DD1] and [BK].