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:
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 = q1q2q3 < 10^5. It contains data for 3132** = 1404 + 1728 fields, which I treated in two intervals of length 510^4. 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:

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 =	9	36	63	117	144	171	225	252	279	333	387	441	468	711	819	1953	Fields
f < 50k	1068	125	35	15	6	7	-	1	-	1	-	1	-	1	-	-	1260
f < 100k	1244	167	61	13	13	9	2	8	1	6	1	1	2	-	1	1	1530
Total	2312	292	96	28	19	16	2	9	1	7	1	2	2	1	1	1	2790

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 =	9	36	63	117	144	171	225	252	279	333	387	441	468	711	819	1953	Fields
f < 50k	1068	125	35	15	6	7	-	1	-	1	-	1	-	1	-	-	1260
f < 100k	1248	167	61	13	13	9	2	8	1	6	1	1	2	-	1	1	1534
Total	2316	292	96	28	19	16	2	9	1	7	1	2	2	1	1	1	2794

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 =	27	108	189	351	675	756	1809	Fields
f < 50k	106	8	2	-	1	1	-	118
f < 100k	148	20	3	1	-	-	1	173
Total	254	28	5	1	1	1	1	291

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 =	27	108	189	351	675	756	1809	Fields
f < 50k	106	8	2	-	1	1	-	118
f < 100k	149	20	3	1	-	-	1	174
Total	255	28	5	1	1	1	1	292

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

h =	81	243	324	567	Fields
f < 50k	25	-	1	-	26
f < 100k	11	3	5	1	20
Total	36	3	6	1	46

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