Computing quadratic function fields with high 3-rank via cubic field tabulation

Abstract

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 \cite {Pieter3} 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 $\mathbb{F}_5, \mathbb{F}_7, \mathbb{F}_11$ and $\mathbb{F}_13$ and $\mathbb{F}_13$. 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.