# 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: Karim.Belabas@math.u-psud.fr aderhem@yahoo.fr danielmayer@algebra.at *

 Quadruplets of Cyclic Cubic Fields (2002/04/17) Dan (02/04/17): In my recent table of cyclic cubic fields the quadruplets sharing a common conductor f were missing up to now. But today I finished the computation of a table of these fields (***) with all 3-prime conductors f = q1*q2*q3 < 10^5. It contains data for 3132 = 1404 + 1728 fields, which I treated in two intervals of length 5*10^4. We must distinguish three cases: (1404 = 1260 + 118 + 26 and 1728 = 1534 + 174 + 20) 1. the fields with 3-class rank rho = 2 for which q1, q2, and q3 are not cubic residues with respect to each other. Here I found the following distribution of class numbers h with respect to conductors f in relative intervals of length 50k = 50000:

h =936631171441712252522793333874414687118191953Fields
f < 50k1068125351567-1-1-1-1--1260
f < 100k124416761131392816112-111530
Total23122929628191629171221112790
Remark:
Four fields are missing from this table: f = 74893, 76921, 81679, 89869.
Here my implementation of Voronoi's algorithm determined the first unit
but ran into an infinite loop for the second unit
(unable to find an association between the Y-branch and the X-chain).
These fields will be investigated separately.
Thus the correct field counts in the last two rows are 1534, 2794.
Dan (02/04/23):
Seeking the second association between the Z-branch and the X-chain
yielded the regulators of the 4 missing fields without problems.
(My first idea to find other generating polynomials without quadratic term
by means of TSCHIRNHAUS transformations turned out to be difficult.)
Using the EULER product method, I got the missing class numbers:
h-- = 9 for f = 74893, where h-+ = 36, h+- = 63, h++ = 9,
h-+ = 9 for f = 76921, where h++ = 333, h+- = 9, h-- = 9,
h+- = 9 for f = 81679, where h-+ = 9, h-- = 9, h++ = 9,
h++ = 9 for f = 89869, where h-+ = 9, h+- = 36, h-- = 36.
Hence, the corrected table is:
h =936631171441712252522793333874414687118191953Fields
f < 50k1068125351567-1-1-1-1--1260
f < 100k124816761131392816112-111534
Total23162929628191629171221112794

 2. the fields with 3-class rank rho = 3 for which two of q1, q2, and q3 are cubic residues with respect to each other:

h =271081893516757561809Fields
f < 50k10682-11-118
f < 100k1482031--1173
Total2542851111291
Remark:
One field is missing from this table: f = 89053.
Here my implementation of Voronoi's algorithm determined the first unit
but ran into an infinite loop for the second unit
(unable to find an association between the Y-branch and the X-chain).
This field will be investigated separately.
Thus the correct field counts in the last two rows are 174, 292.
Dan (02/04/23):
Seeking the second association between the Z-branch and the X-chain
yielded the regulator of the missing field without problems.
Using the EULER product method, I got the missing class number:
h+- = 27 for f = 89053, where h-- = 9, h++ = 9, h+- = 27.
Hence, the corrected table is:
h =271081893516757561809Fields
f < 50k10682-11-118
f < 100k1492031--1174
Total2552851111292

 3. the fields with 3-class rank rho = 4, for which q1, q2, and q3 are mutual cubic residues:

h =81243324567Fields
f < 50k25-1-26
f < 100k1135120
Total3636146