Explicit solution by radicals, gonal maps and plane models of algebraic curves of genus 5 or 6

We give explicit computational algorithms to construct minimal degree (always ⩽4) ramified covers of P1 for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the g⩽4 case) on constructing radical parametrisations of arbitrary genus g curves. Zariski showed that this is impossible for the general curve of genus ⩾7. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.