Abstract
We give a general method for tabulating all cubic function fields over 
whose discriminant D has odd degree, or even degree such that the leading coefficient of − 3D is a non-square in 
, up to a given bound on \(|D| = q^{\deg(D)}\). The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields. We present numerical data for cubic function fields over 
and over 
with \(\deg(D) \leq 7\) and \(\deg(D)\) odd in both cases.
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References
Artin, E.: Quadratische Körper im Gebiete der höheren Kongruenzen I. Math. Zeitschrift 19, 153–206 (1924)
Belabas, K.: A fast algorithm to compute cubic fields. Math. Comp. 66(219), 1213–1237 (1997)
Belabas, K.: On quadratic fields with large 3-rank. Math. Comp. 73(248), 2061–2074 (2004)
Cohen, H.: Advanced Topics in Computational Number Theory. Springer, New York (2000)
Cremona, J.E.: Reduction of binary cubic and quartic forms. LMS J. Comput. Math. 2, 62–92 (1999)
Datskovsky, B., Wright, D.J.: Density of discriminants of cubic extensions. J. reine angew. Math. 386, 116–138 (1988)
Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields I. Bull. London Math. Soc. 1, 345–348 (1969)
Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields II. Proc. Royal Soc. London A 322, 405–420 (1971)
Rosen, M.: Number Theory in Function Fields. Springer, New York (2002)
Rozenhart, P.: Fast Tabulation of Cubic Function Fields. PhD Thesis, University of Calgary (in progress)
Shoup, V.: NTL: A Library for Doing Number Theory. Software (2001), http://www.shoup.net/ntl
Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, New York (1993)
Taniguchi, T.: Distributions of discriminants of cubic algebras (preprint, 2006), http://arxiv.org/abs/math.NT/0606109
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Rozenhart, P., Scheidler, R. (2008). Tabulation of Cubic Function Fields with Imaginary and Unusual Hessian. In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_24
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DOI: https://doi.org/10.1007/978-3-540-79456-1_24
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