A new curve algebraically but not rationally uniformized by radicals

We give a new example of a curve C algebraically, but not rationally, uniformized by radicals. This means that C has no map onto P1 with solvable Galois group, while there exists a curve C′ that maps onto C and has a finite morphism to P1 with solvable Galois group. We construct such a curve C of genus 9 in the second symmetric product of a general curve of genus 2. It is also an example of a genus 9 curve that does not satisfy condition S(4,2,9) of Abramovich and Harris