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Mathematics of Computation

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On quadratic fields with large 3-rank

Author: Karim Belabas
Translated by:
Journal: Math. Comp. 73 (2004), 2061-2074
MSC (2000): Primary 11R11, 11R16, 11Y40
Published electronically: January 30, 2004
MathSciNet review: 2059751
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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 $O(X)$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 $5$.

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Additional Information

Karim Belabas
Affiliation: Université Paris-Sud, Département de Mathématiques (bât. 425), F-91405 Orsay, France

Keywords: Cubic fields, quadratic fields, 3-rank
Received by editor(s): April 8, 2002
Received by editor(s) in revised form: May 3, 2003
Published electronically: January 30, 2004
Article copyright: © Copyright 2004 American Mathematical Society