# 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 *

 The Leading Table of Cyclic Cubic Fields (2002/04/08) Dan (02/04/08): Now I have completed the most extensive tables of cyclic cubic fields of all times: (*) with all prime conductors f = q < 10^5, (**) with all 2-prime conductors f = q1*q2 < 10^5. They contain data for 12511 = 4785 + 6910 + 816 fields. The method of construction has been explained in a previous communication. This extends the computational results of ENNOLA and TURUNEN, which were bounded by f < 16000. 1. In the first table (*) I found the following distribution of class numbers h among these fields with 3-class rank rho = 0 with respect to conductors f in relative intervals of length 10k = 10000:

h =14713161925283137434952616467737691100103109111?112121124127128133149169171172175193208211217223229343348349363364510532688Fields
f < 10k50661174471531-2----1-------------------------------612
f < 20k42057113312-1122321----1-2----1---------------------513
f < 30k38959137-3-321---211-2--1--1------------------1-----486
f < 40k3756912225131--122-111----1-------------------------479
f < 50k3736310523-21-------2-11---1----1-1-------1---------467
f < 60k374561325----1----1---11----1---1-1-1---------------458
f < 70k36135138-1-12-----11---1-------1---1-11--1----------429
f < 80k378481242112--------1-1----------1----------11-----1454
f < 90k36049183-1--1---1--------1---1--1------11--------1--439
f < 100k3476111571-2----21-1-1----------1------1---1---11-1-445
Total3883558130432523518114258744543413121111412111121111111111114782
Remarks:
1. Three fields are missing from this table: f = 77587, 83383, 96847.
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 four rows are 455, 440, 446, 4785.
2. The class number h = 111 = 3*37 is a contradiction to 3-rank 0.
Here the Euler product must obviously be evaluated by using more primes.
This field (f = 31513) will be investigated separately.
Dan (02/04/23):
Seeking the second association between the Z-branch and the X-chain
yielded the regulators of the 3 missing fields without problems.
(My first idea to find other generating polynomials without quadratic term
by means of TSCHIRNHAUS transformations failed.)
Using the EULER product method, I got the missing class numbers:
h = 1 for f = 77587,
h = 4 for f = 83383,
h = 1 for f = 96847,
and the corrected class number (with primes up to 39971 instead of 9733):
h = 112 = 2^4*7 for f = 31513.
Hence, the corrected table is:
h =14713161925283137434952616467737691100103109112121124127128133149169171172175193208211217223229343348349363364510532688Fields
f < 10k50661174471531-2----1------------------------------612
f < 20k42057113312-1122321----1-2---1---------------------513
f < 30k38959137-3-321---211-2--1-1------------------1-----486
f < 40k3756912225131--122-111----1------------------------479
f < 50k3736310523-21-------2-11--1----1-1-------1---------467
f < 60k374561325----1----1---11---1---1-1-1---------------458
f < 70k36135138-1-12-----11---1------1---1-11--1----------429
f < 80k379481242112--------1-1---------1----------11-----1455
f < 90k36050183-1--1---1--------1--1--1------11--------1--440
f < 100k3486111571-2----21-1-1---------1------1---1---11-1-446
Total388555913043252351811425874454341331111412111121111111111114785

 2. In the second table (**) we must distinguish two cases: a) the fields with 3-class rank rho = 1 for which q1 and q2 are not cubic residues with respect to each other:

h =31221394857758493111129147156183192201219228237273300327336375444453525588591597624669Fields
f < 10k590792754322---------2--------------714
f < 20k60570211223212111---------1----------722
f < 30k5699117432-3-4-11----1--1-1---------698
f < 40k56971211073-1-21231---2-1------------694
f < 50k554801659411521--11121---1--1-------686
f < 60k551752154517---23--112-11-1--1--1---684
f < 70k586622896312---1211--11-1--1-----1-1708
f < 80k5487724166142---111112--1------------686
f < 90k559672421142-11-1-121-11-----1-1-----680
f < 100k5196521261-8--2131--2-----1----1--1-634
Total5650737220705829132781051013654710233231211111116906
Remarks:
1. Four fields are missing from this table: f = 90961, 93367, 93733, 97183.
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 638, 6910.
2. Further the field f = 51763 appears incorrectly in the next table
and is missing here, since its class number was calculated as h = 414 = 3^2*2*23.
Dan (02/04/24):
Seeking the second association between the Z-branch and the X-chain
(resp. between the Z-branch and the Y-chain for the particularly hard-boiled f = 97183)
yielded the regulators of the 4 missing fields without problems.
Using the EULER product method, I got the missing class numbers:
h- = 3 for f = 90961, where h+ = 12,
h+ = 12 for f = 93367, where h- = 3,
h- = 3 for f = 93733, where h+ = 183,
h- = 3 for f = 97183, where h+ = 3,
and the corrected class number (with primes up to 27449 instead of 12553):
h- = 417 = 3*139 for f = 51763, where h+ = 3.
Hence, the corrected table is:
h =31221394857758493111129147156183192201219228237273300327336375417444453525588591597624669Fields
f < 10k590792754322---------2---------------714
f < 20k60570211223212111---------1-----------722
f < 30k5699117432-3-4-11----1--1-1----------698
f < 40k56971211073-1-21231---2-1-------------694
f < 50k554801659411521--11121---1---1-------686
f < 60k551752154517---23--112-11-1-1-1--1---684
f < 70k586622896312---1211--11-1--1------1-1708
f < 80k5487724166142---111112--1-------------686
f < 90k559672421142-11-1-121-11------1-1-----680
f < 100k5226621261-8--2131--2-----1-----1--1-638
Total56537382207058291327810510136547102332311211111116910

 b) the fields with 3-class rank rho = 2, for which q1 and q2 are mutual cubic residues:

h =927366381108117144171189225243252279333351414?Fields
f < 10k60454--1----------74
f < 20k55972-----1-------74
f < 30k68773-2---1-------88
f < 40k53732-3--11-11----72
f < 50k471067312113-111---84
f < 60k601443-11--11-1---(1)86
f < 70k63483152-211-1--1-92
f < 80k4996--21-111-2----72
f < 90k70753--32-11------92
f < 100k5588311------1-1--78
Total580795930515103510427111(1)812
Remarks:
1. Four fields are missing from this table: f = 94087, 96091, 98557 (twice).
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 82, 816.
2. The class number h = 414 = 3^2*2*23 is impossible,
since the class group cannot have cyclic subgroups of order 2 and 23.
Here the Euler product must obviously be evaluated by using more primes.
This field (f = 51763 = 37*1399) with 3-rank 1(!) will be investigated separately.
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.
Using the EULER product method, I got the missing class numbers:
h+ = 9 for f = 94087, where h- = 9,
h- = 36 for f = 96091, where h+ = 9,
h- = 36 and h+ = 9 for f = 98557,
and the corrected class number (with primes up to 27449 instead of 12553):
h- = 417 = 3*139 for f = 51763, where h+ = 3.
Hence, the corrected table is:
h =927366381108117144171189225243252279333351Fields
f < 10k60454--1---------74
f < 20k55972-----1------74
f < 30k68773-2---1------88
f < 40k53732-3--11-11---72
f < 50k471067312113-111--84
f < 60k601443-11--11-1---86
f < 70k63483152-211-1--192
f < 80k4996--21-111-2---72
f < 90k70753--32-11-----92
f < 100k57810311------1-1-82
Total582796130515103510427111816