Fast Tabulation of Cubic Function Fields with Applications

We give a general method for tabulating all cubic function fields whose discriminant D has odd degree, or even degree such that the leading coefficient of -3D is a non-square in the underlying finite field, up to a given bound on |D|=q^(deg(D)). We highlight the main theoretical ingredients required and present numerical data for the cubic function fields we tabulate. In addition, we modify our tabulation algorithm to compute 3-ranks of quadratic function fields by way of a generalization of a theorem of Hasse. The algorithm, whose complexity is linear in the number of reduced binary cubic forms up to some upper bound X on |D|, is described and numerical results are given.