  We develop in this article an algorithm that, given a projective curve $C$,
computes a \textit{gonal map}, that is, a finite morphism from $C$ to the
projective line of minimal degree. Our method is based on the computation of
scrollar syzygies of canonical curves. We develop an improved version of our
algorithm for curves with a unique gonal map and we discuss a characterization
of such curves in terms of Betti numbers. Finally, we derive an efficient
algorithm for radical parametrization of curves of gonality $\le 4$.
