We give an explicit description of the behaviour of the root number in the family given by twists of an elliptic curve E by the rational values of a polynomial f(T). In particular, we give a criterion (on f depending on E) for the family to have a constant root number over Q. This completes a work of Rohrlich: we detail the behaviour of the root number when E has bad reduction over Qab and we treat the cases j(E)=0,1728 which were not considered previously