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# Computing quadratic function fields with high 3-rank via cubic field tabulation

1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : We present recent results on the computation of quadratic function fields with high 3-rank. Using a generalization of a method of Belabas on cubic field tabulation and a theorem of Hasse, we compute quadratic function fields with 3-rank $\geq 1$, of imaginary or unusual discriminant $D$, for a fixed $|D| = q^{\deg(D)}$. We present numerical data for quadratic function fields over $\mathbb{F}_{5}, \mathbb{F}_{7}, \mathbb{F}_{11}$ and $\mathbb{F}_{13}$ with $\deg(D) \leq 11$. Our algorithm produces quadratic function fields of minimal genus for any given 3-rank. Our numerical data mostly agrees with the Friedman-Washington heuristics for quadratic function fields over the finite field $\mathbb{F}_{q}$ where $q \equiv -1 \pmod{3}$. The data for quadratic function fields over the finite field $\mathbb{F}_{q}$ where $q \equiv 1 \pmod{3}$ does not agree closely with Friedman-Washington, but does agree more closely with some recent conjectures of Malle.
Document type :
Journal articles
Complete list of metadatas

https://hal.inria.fr/inria-00462008
Contributor : Pieter Rozenhart <>
Submitted on : Monday, March 8, 2010 - 11:02:41 AM
Last modification on : Friday, December 21, 2018 - 2:52:48 PM

### Identifiers

• HAL Id : inria-00462008, version 1
• ARXIV : 1003.1287

### Citation

Pieter Rozenhart, Michael Jacobson Jr., Renate Scheidler. Computing quadratic function fields with high 3-rank via cubic field tabulation. Rocky Mountain Journal of Mathematics, Rocky Mountain Mathematics Consortium, 2015, 45 (6), pp.1985-2022. ⟨inria-00462008⟩

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