Numerical results on class groups of imaginary quadratic fields

Abstract

Using techniques described in [3], we have computed the class number and class group structure of all imaginary quadratic fields with discriminant Δ for 0 < |Δ| < 1011. A novel verification algorithm based on the Eichler Selberg Trace Formula [15] was used to ensure that the correctness of our results does not rely on any unproved hypothesis. We present the results of our computations, and remark on specific evidence that was found pertaining to a number of heuristics. In particular, we present data which supports some of the Cohen-Lenstra heuristics [8], Littlewood's bounds on L(1, Χ) [14], and Bach's bound on the maximum norm of the prime ideals required to generate the class group [1]. © Springer-Verlag Berlin Heidelberg 2006.