Regulator and Class Group Tabulation in Real Quadratic Number Fields

Abstract

In this work, the regulator R_∆ and class group structure Cl_∆ for all real quadratic fields Q(√∆), ∆ ≤ 1.1 × 2^(40) are computed conditionally. The results are then verified unconditionally correct for ∆ ≤ 1.1 × 10^(12) (that is, ∆ ≤ 2^(40)). This is achieved, in part, by adapting previous methods for computation in imaginary quadratic fields and proving certain time complexity results on them. A statistical aggregate on the tabulated results is made and compared to several longstanding conjectures and conditional theorems. Several fields are also found and documented to have invariants with novel properties and class group structures.