Abstract: The Picard number of a rational surface equipped with a relatively minimal trigonal fibration is bounded in terms of the genus $g$ of a general fibre. When the Picard number attains the maximum for $g\geq3$, we describe such a fibred surface whose Mordell-Weil group is trivial. This is a joint work with Cheng Gong (Soochow Univ.) and Jun Lu (East China Normal Univ.).