  We describe and give computational results of a procedure to compute the
divisor class number and regulator of most purely cubic function fields of unit
rank 2. Our implementation is an improvement to Pollard's Kangaroo method in
infrastructures, using distribution results of class numbers as well as
information on the congruence class of the divisor class number, and an
adaptation that efficiently navigates these torus-shaped infrastructures.
Moreover, this is the first time that an efficient "square-root" algorithm has
been applied to the infrastructure of a global field of unit rank 2. With the
exception of certain function fields defined by Picard curves, our examples are
the largest known divisor class numbers and regulators ever computed for a
function field of genus 3.
