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On these pages, we present most recent results of our joint research, directly from the lab. 
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 12131237 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), 831847 and S55S58 
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Email addresses:
Karim.Belabas@math.upsud.fr aderhem@yahoo.fr danielmayer@algebra.at 
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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 2prime 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 3class rank rho = 0 with respect to conductors f in relative intervals of length 10k = 10000: 


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 Ybranch and the Xchain). 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 3rank 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 Zbranch and the Xchain 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: 


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


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 Ybranch and the Xchain). 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 Zbranch and the Xchain (resp. between the Zbranch and the Ychain for the particularly hardboiled 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: 


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


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 Ybranch and the Xchain). 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 3rank 1(!) will be investigated separately. 

Dan (02/04/23):
Seeking the second association between the Zbranch and the Xchain 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: 


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