Joint research 2002

of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer

On these pages, we present most recent results of our joint research, directly from the lab.
Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237
A. Derhem, Capitulation dans les extensions quadratiques de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Québec, 1988
D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58
E-mail addresses:

Octuplets of Cyclic Cubic Fields (2002/04/20)
Dan (02/04/20):
This is the last notch in my table of cyclic cubic fields:
the octuplets sharing a common conductor f with four prime divisors.
The regulators, being rather small, went through without any difficulty.

However, the identification of the individual members of such an 8-family
with the aid of splitting primes turned out to be tedious.
Whereas in all former cases of conductors with up to three prime divisors
any field could be characterized by some prime that splits only in this single field,
we have now the phenomenon that any prime splits at least in 2 members of an octuplet.

Another complication arose from the extremely slow convergence of the EULER product.
I had to coarsen the admissible interval of fluctuation
from [h-0.1,h+0.1] to [h-0.45,h+0.45],
since first the limit was clear in each case and
second I didn't want to use primes bigger than 10^6.
Thus, I finally give a table of these arduous fields

(****) with all 4-prime conductors f = q1*q2*q3*q4 < 10^5.

It contains data for 208 fields,
which were treated at once in an absolute interval of length 10^5.

We must distinguish four cases:

1. the fields with 3-class rank rho = 3
for which q1, q2, q3, and q4 are not cubic residues with respect to each other.
Here I found the following distribution of class numbers h:

h =27108189432513Fields
f < 100k154211111188

2.,3.,4. the fields with 3-class ranks rho = 4,5,6
for which some or all of q1, q2, q3, and q4 are cubic residues with respect to each other.

h =81567Fields
f < 100k11112

h =243486Fields
f < 100k617

h =729Fields
f < 100k11

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