On quadratic fields with large 3-rank
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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 $5$.References
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Additional Information
- Karim Belabas
- Affiliation: Université Paris-Sud, Département de Mathématiques (bât. 425), F-91405 Orsay, France
- Email: Karim.Belabas@math.u-psud.fr
- Received by editor(s): April 8, 2002
- Received by editor(s) in revised form: May 3, 2003
- Published electronically: January 30, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 2061-2074
- MSC (2000): Primary 11R11, 11R16, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-04-01632-1
- MathSciNet review: 2059751