Tabulation of cubic function fields via polynomial binary cubic forms

We present a method for tabulating all cubic function fields over Fq(t) whose discriminant D has either odd degree or even degree and the leading coefficient of −3D is a non-square in Fq , up to a given bound B on deg(D). Our method is based on a generalization of Belabas’ method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires O(B4qB) field operations as B → ∞. The algorithm, examples and numerical data for q = 5, 7, 11, 13 are included.