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We present computational results on the divisor class number and theregulator of a cubic function field over a large base field. The underlyingmethod is based on approximations of the Euler product representation of thezeta function of such a field. We give details on the implementation for purelycubic function fields of signatures $(3,1)$ and $(1, 1; 1, 2)$, operating inthe ideal class group and infrastructure of the function field, respectively.Our implementation provides numerical evidence of the computationaleffectiveness of this algorithm. With the exception of special cases, such aspurely cubic function fields defined by superelliptic curves, the examplesprovided are the largest divisor class numbers and regulators ever computed fora cubic function field over a large prime field. The ideas underlying theoptimization of the class number algorithm can in turn be used to analyze thedistribution of the zeros of the function field's zeta function. We provide avariety of data on a certain distribution of the divisor class number thatverify heuristics by Katz and Sarnak on the distribution of the zeroes of thezeta function.

Class number and regulator computation in cubic function fields