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 $O(X^{3/4})$, or more generally in time $O(X + X^{7/4} / M)$ and space $O(M + X^{1/2})$ 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 $r_3(\Delta) \geq 4$, and prove that $\mathbb{Q} (\sqrt{-5393946914743})$ is the smallest imaginary quadratic field with $3$-rank equal to $5$.