A curve algebraically but not rationally uniformized by radicals

Zariski proved the general complex projective curve of genus g>6 is not rationally uniformized by radicals, that is, admits no map to whose Galois group is solvable. We give an example of a genus seven complex projective curve Z that is not rationally uniformized by radicals, but such that there is a finite covering Z′→Z with Z′ rationally uniformized by radicals. The curve providing the example appears in a paper by Debarre and Fahlaoui where a construction is given to show the Brill Noether loci Wd(C) in the Jacobian of a curve C may contain translates of abelian subvarieties not arising from maps from C to other curves.