Extending previous work of H. W. Lenstra, Jr. and the first author, we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four (given the signature and the Galois group of the Galois closure). The theoretical justifications for these conjectures will appear elsewhere, but the agreement with the existing tables is quite good. 1. Introduction and Notations. In (3), H. W. Lenstra, Jr., and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree. In a forthcoming paper (4) we shall show that this technique can be extended to a much wider class of number fields, and also to relative extensions. The aim of the present paper is to rapidly make available the numerical conjec- tures obtained, for people not really interested in our heuristic reasoning or not wanting to wait for (4) to appear. Hence, apart from a total lack of justifications for the conjectures that we present, this paper is essentially self-contained. The plan is as follows. In the rest of this section we present the notations used in the sequel. Some of them being nonstandard (and in general differing from the notations of (3)), we urge the reader to read the notations carefully before applying the conjectures. In the next section we present templates for the subsequent conjectures, and then the conjectures themselves, illustrated by numerical examples, first for their own sake, and second as a double check for the reader to understand the templates. These conjectures are given for all types of fields of degree less than or equal to four. In the final section we comment on the consistency of the conjectures with existing tables (which is quite good). Combinatorial Notations: * If X is a set, {XI denotes its cardinality.
[1]
Marie Nicole Gras,et al.
Méthodes et algorithmes pour le calcul numérique du nombre de classes et de unités des extensions cubiques cycliques de Q.
,
1975
.
[2]
Sirpa Mäki,et al.
The determination of units in real cyclic sextic fields
,
1980
.
[3]
G. Frey,et al.
Computation of the 2-rank of pure cubic fields
,
1978
.
[4]
M. Gras.
Classes et unités des extensions cycliques réelles de degré 4 de ${\bf Q}$
,
1979
.
[5]
Veikko Ennola,et al.
On cyclic cubic fields
,
1985
.
[6]
H. C. Williams,et al.
A note on class-number one in certain real quadratic and pure cubic fields
,
1986
.
[7]
Veikko Ennola,et al.
On real cyclic sextic fields
,
1985
.
[8]
Christopher Hooley.
On the Pellian equation and the class number of indefinite binary quadratic forms.
,
1984
.
[9]
Henri Cohen,et al.
Heuristics on class groups of number fields
,
1984
.
[10]
I. Angell.
A Table of Complex Cubic Fields
,
1973
.
[11]
H. C. Williams,et al.
A note on class-number one in pure cubic fields
,
1979
.
[12]
Marie Nicole Gras,et al.
Methodes et algorithmes pour le calcul numerique du nombre de classes et des unites des extensions cubiques cycliques de Q
,
1975
.
[13]
Veikko Ennola,et al.
On Totally Real Cubic Fields
,
1985
.
[14]
Duncan A. Buell.
The expectation of success using a Monte Carlo factoring method—some statistics on quadratic class numbers
,
1984
.