N-covers of hyperelliptic curves

For a hyperelliptic curve e of genus g with a divisor class of order n = g + 1, we shall consider an associated covering collection of curves D-delta, each of genus g(2). We describe, up to isogeny, the Jacobian of each D-delta via a map from D-delta to e, and two independent maps from D-delta to a curve of genus g(g - 1)/2. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all Q-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.