We give explicit computational algorithms to construct minimal degree (always
≤4) ramified covers of \Prj^1 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.Comment: v3: full version of the pape