The hyperelliptic curve C of genus 2 given by
y^2=x^5-x^3+x
has class group isomorphic to Z/8*Z/8. The points \infty, (0,0), (4,3) and (4,2) on C map to (0,4), 0, (0,5) and (0,3) in the class group, hence to a subgroup of index 8. By class field theory, C has an unramified cover of degree 8 in which these points split completely. This cover therefore has genus 1+8*(2-1)=9 and at least 8*4=32 rational points (in fact exactly 32 since this was already known to be an upper bound).