On quadratic fields with large 3-rank

Abstract

Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem then analyze and improve the table-building algorithm. It computes the multiplicities of the O ( X ) O(X) general cubic discriminants (real or imaginary) up to X X in time O ( X ) O(X) and space O ( X 3 / 4 ) O(X^{3/4}) , or more generally in time O ( X + X 7 / 4 / M ) O(X + X^{7/4} / M) and space O ( M + X 1 / 2 ) O(M + X^{1/2}) for a freely chosen positive M M . A variant computes the 3 3 -ranks of all quadratic fields of discriminant up to X X with the same time complexity, but using only M + O ( 1 ) M + O(1) units of storage. As an application we obtain the first 1618 1618 real quadratic fields with r 3 ( Δ ) ≥ 4 r_3(\Delta ) \geq 4 , and prove that Q ( − 5393946914743 ) \mathbb {Q}(\sqrt {-5393946914743}) is the smallest imaginary quadratic field with 3 3 -rank equal to 5 5 .