  We explore parameterizations by radicals of low genera algebraic curves. We
prove that for $q$ a prime power that is large enough and prime to $6$, a fixed
positive proportion of all genus 2 curves over the field with $q$ elements can
be parameterized by $3$-radicals. This results in the existence of a
deterministic encoding into these curves when $q$ is congruent to $2$ modulo
$3$. We extend this construction to parameterizations by $\ell$-radicals for
small odd integers $\ell$, and make it explicit for $\ell=5$.
