COMPUTING QUADRATIC FUNCTION FIELDS WITH HIGH 3-RANK VIA CUBIC FIELD TABULATION

In this paper, we present extensive numerical data on quadratic function fields with non-zero 3-rank. We use a function field adaptation of a method due to Belabas for finding quadratic number fields of high 3-rank. Our algorithm relies on previous work for tabulating cubic function fields of bounded discriminant [28] but includes a significant novel improvement when the discriminants are imaginary. We provide numerical data for discriminant degree up to 11 over the finite fields 75,77,711 and 713. In addition to presenting new examples of fields of minimal discriminant degree with a given 3-rank, we compare our data with a variety of heuristics on the density of such fields with a given 3-rank, which in most cases supports their validity.