  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 $\mathbb{Q}$. This completes
a work of Rohrlich: we detail the behaviour of the root number when $E$ has bad
reduction over $\mathbb{Q}^{ab}$ and we treat the cases $j(E)=0,1728$ which
were not considered previously.
