Heuristics on class groups: some good primes are not too good. (English) Zbl 0827.11067

The authors reexamine the prediction of their heuristics for class groups of number fields [Math. Comput. 48, 123–137 (1987; Zbl 0627.12006)], in relation to more recent computations of class groups of complex cubic fields by G. Fung and H. C. Williams [Math. Comput. 55, 313–325 (1990; Zbl 0705.11063)]. These computations bring to light some apparent discrepancies with the predicted proportions of complex cubic fields for which the prime to 3 part of the class group has certain properties. The explanation that the authors propose for these discrepancies is that the prime 2 must be handled in a different way, rather than being treated as a “good” prime as in their prior analysis.


11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions
11Y40 Algebraic number theory computations
Full Text: DOI


[1] H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33 – 62. · doi:10.1007/BFb0099440
[2] H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), no. 177, 123 – 137. · Zbl 0627.12006
[3] Henri Cohen and Jacques Martinet, Étude heuristique des groupes de classes des corps de nombres, J. Reine Angew. Math. 404 (1990), 39 – 76 (French). · Zbl 0699.12016
[4] H. Eisenbeis, G. Frey, and B. Ommerborn, Computation of the 2-rank of pure cubic fields, Math. Comp. 32 (1978), no. 142, 559 – 569. · Zbl 0385.12001
[5] D. Ford, Enumeration of totally complex quartic fields of small discriminant, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 129 – 138. · Zbl 0729.11051
[6] Gilbert W. Fung and H. C. Williams, On the computation of a table of complex cubic fields with discriminant \?>-10\(^{6}\), Math. Comp. 55 (1990), no. 191, 313 – 325. · Zbl 0705.11063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.