On quadratic fields with large 3-rank

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) general cubic discriminants (real or imaginary) up to X in time O(X) and space O(X3/4), or more generally in time O(X+X7/4/M) and space O(M +X1/2) for a freely chosen positive M . A variant computes the 3-ranks of all quadratic fields of discriminant up to X with the same time complexity, but using only M + O(1) units of storage. As an application we obtain the first 1618 real quadratic fields with r3(∆) ≥ 4, and prove that Q( √ −5393946914743) is the smallest imaginary quadratic field with 3-rank equal to 5.