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2015 Computing quadratic function fields with high 3-rank via cubic field tabulation
P. Rozenhart, M.J. Jacobson, Jr, R. Scheidler
Rocky Mountain J. Math. 45(6): 1985-2022 (2015). DOI: 10.1216/RMJ-2015-45-6-1985

Abstract

In this paper, we present extensive numerical data on quadratic function fields with non-zero 3-rank. We use a function field adaptation of a method due to Belabas for finding quadratic number fields of high 3-rank. Our algorithm relies on previous work for tabulating cubic function fields of bounded discriminant \cite {Pieter3} but includes a significant novel improvement when the discriminants are imaginary. We provide numerical data for discriminant degree up to 11 over the finite fields $\mathbb{F}_5, \mathbb{F}_7, \mathbb{F}_11$ and $\mathbb{F}_13$ and $\mathbb{F}_13$. In addition to presenting new examples of fields of minimal discriminant degree with a given 3-rank, we compare our data with a variety of heuristics on the density of such fields with a given 3-rank, which in most cases supports their validity.

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P. Rozenhart. M.J. Jacobson, Jr. R. Scheidler. "Computing quadratic function fields with high 3-rank via cubic field tabulation." Rocky Mountain J. Math. 45 (6) 1985 - 2022, 2015. https://doi.org/10.1216/RMJ-2015-45-6-1985

Information

Published: 2015
First available in Project Euclid: 14 March 2016

zbMATH: 1350.11095
MathSciNet: MR3473164
Digital Object Identifier: 10.1216/RMJ-2015-45-6-1985

Subjects:
Primary: 11R11, 11R58, 11R65, 11Y40

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium

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Vol.45 • No. 6 • 2015
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