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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.

MSC:

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

[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
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