# 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 non ramifiées 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 * Web master's e-mail address: contact@algebra.at *

 Unexplored Pure Cubic Fields with 3-class rank r >= 2 (2002/10/26) Dan (02/10/26): Now, after the previous four reports on 3-class numbers of pure cubic fields K = Q( R1/3 ), what remains to be done in the next instance? Let us estimate the 3-class rank rK with the aid of inequalities rmin <= rK <= rmax, communicated by Barrucand, Williams, and Baniuk in [0] on the basis of results by Kobayashi and Gerth. In the following table, let the conductor f of the normal closure N of K over its quadratic subfield k be characterized by the following counters: t = #{ q prime | q divides f } ... total number of prime divisors, n = #{ q prime | q divides f, q incongruent 1 (mod 3) } ... not split prime divisors, s = #{ q prime | q divides f, q congruent 1 (mod 3) } ... split prime divisors, u = #{ q prime | q divides f, q congruent 1,8 (mod 9) } ... free prime divisors, v = #{ q prime | q divides f, q incongruent 1,8 (mod 9) } ... restrictive prime divisors, e = v3(f) ... exact 3-exponent of f, that characterizes Dedekind's types 1A (e = 2), 1B (e = 1), and 2 (e = 0), m = m(f) ... multiplicity of the conductor f (number of non-isomorphic fields sharing f), t0 = #{ q prime, q != 3 | q divides f }, v0 = #{ q prime, q != 3 | q divides f, q incongruent 1,8 (mod 9) }, d = 0 if v0 = 0 and d = 1 if v0 >= 1, r = t - 1 - d, rmin = max(s,r), rmax = s + r, # ... the number of occurrences for R < 106.

t __ e n s u v __ m __ t0 v0 d r rmin rmax __ f R conditions # reference

1 2 1 0 0 1 1 0 0 0 0 0 0 9 3 1 [1]
.
1 0 1 0 1 0 1 1 0 0 0 0 0 q q q = 8 (9) 13099 [1]
1 0 0 1 1 0 1 1 0 0 0 1 1 p p p = 1 (9) 13063 [2],Type 1

2 1 2 0 0 2 1 1 1 1 0 0 0 3q q q = 2,5 (9) 26167 [1]
2 1 1 1 0 2 1 1 1 1 0 1 1 3p p p = 4,7 (9) 26168 [2],Type 1
.
2 2 2 0 0 2 2 1 1 1 0 0 0 9q 3q,3q2,9q q = 2,5 (9) 13098 [1]
2 2 2 0 1 1 2 1 0 0 1 1 1 9q 3q,9q q = 8 (9) 6538 [2],Type 2
2 2 1 1 0 2 2 1 1 1 0 1 1 9p 3p,9p p = 4,7 (9) 13048 [2],Type 1
2 2 1 1 1 1 2 1 0 0 1 1 2 9p 3p,9p p = 1 (9) 6523 [3],Type 3
.
2 0 2 0 0 2 1 2 2 1 0 0 0 q1q2 q1q2,q12q2 q1,q2 = 2,5 (9) 21520 [1]
2 0 2 0 2 0 2 2 0 0 1 1 1 q1q2 q1q2,q12q2 q1,q2 = 8 (9) 3007 [2],Type 2
2 0 1 1 0 2 1 2 2 1 0 1 1 pq pq,p2q,pq2 p = 4,7, q = 2,5 (9) 29615 [2],Type 1
2 0 1 1 2 0 2 2 0 0 1 1 2 pq pq,p2q,pq2 p = 1, q = 8 (9) 5948 [3],Type 3
2 0 0 2 0 2 1 2 2 1 0 2 2 p1p2 p1p2,p12p2 p1,p2 = 4,7 (9) 8210
2 0 0 2 2 0 2 2 0 0 1 2 3 p1p2 p1p2,p12p2 p1,p2 = 1 (9) 2913

3 1 3 0 0 3 1 2 2 1 1 1 1 3q1q2 q1q2,q12q2 q1,q2 = 2,5 (9) 21460 [2],Type 2
3 1 3 0 1 2 2 2 1 1 1 1 1 3q1q2 q1q2,q12q2,q1q22 q1 = 2,5 (9), q2 = 8 (9) 27510 [2],Type 2
3 1 2 1 0 3 1 2 2 1 1 1 2 3pq pq,p2q,pq2 p = 4,7, q = 2,5 (9) 29714 [3],Type 3
3 1 2 1 1 2 2 2 1 1 1 1 2 3pq pq,p2q,pq2 p = 4,7, q = 8 (9) 14286 [3],Type 3
3 1 2 1 1 2 2 2 1 1 1 1 2 3pq pq,p2q,pq2 p = 1, q = 2,5 (9) 27285 [3],Type 3
3 1 1 2 0 3 1 2 2 1 1 2 3 3p1p2 p1p2,p12p2 p1,p2 = 4,7 (9) 8170
3 1 1 2 1 2 2 2 1 1 1 2 3 3p1p2 p1p2,p12p2,p1p22 p1 = 4,7 (9), p2 = 1 (9) 14026
.
3 2 3 0 0 3 4 2 2 1 1 1 1 9q1q2 3xq1yq2z q1,q2 = 2,5 (9) 20999 [2],Type 2
3 2 3 0 1 2 4 2 1 1 1 1 1 9q1q2 3xq1yq2z q1 = 2,5 (9), q2 = 8 (9) 13171 [2],Type 2
3 2 3 0 2 1 4 2 0 0 2 2 2 9q1q2 3xq1yq2z q1,q2 = 8 (9)
3 2 2 1 0 3 4 2 2 1 1 1 2 9pq 3xpyqz p = 4,7, q = 2,5 (9) 28491 [3],Type 3
3 2 2 1 1 2 4 2 1 1 1 1 2 9pq 3xpyqz p = 4,7, q = 8 (9) 6478 [3],Type 3
3 2 2 1 1 2 4 2 1 1 1 1 2 9pq 3xpyqz p = 1, q = 2,5 (9) 13002 [3],Type 3
3 2 2 1 2 1 4 2 0 0 2 2 3 9pq 3xpyqz p = 1, q = 8 (9)
3 2 1 2 0 3 4 2 2 1 1 2 3 9p1p2 3xp1yp2z p1,p2 = 4,7 (9) 7523
3 2 1 2 1 2 4 2 1 1 1 2 3 9p1p2 3xp1yp2z p1 = 4,7 (9), p2 = 1 (9) 6342
3 2 1 2 2 1 4 2 0 0 2 2 4 9p1p2 3xp1yp2z p1,p2 = 1 (9)
.
3 0 3 0 0 3 1 3 3 1 1 1 1 q1q2q3 q1xq2yq3z q1,q2,q3 = 2,5 (9) 5249 [2],Type 2
3 0 3 0 1 2 2 3 2 1 1 1 1 q1q2q3 q1xq2yq3z q1,q2 = 2,5 (9), q3 = 8 (9) 9661 [2],Type 2
3 0 3 0 3 0 4 3 0 0 2 2 2 q1q2q3 q1xq2yq3z q1,q2,q3 = 8 (9)
3 0 2 1 0 3 1 3 3 1 1 1 2 pq1q2 pxq1yq2z p = 4,7, q1,q2 = 2,5 (9) 10813 [3],Type 3
3 0 2 1 1 2 2 3 2 1 1 1 2 pq1q2 pxq1yq2z p = 4,7, q1 = 2,5 (9), q2 = 8 (9) 10292 [3],Type 3
3 0 2 1 1 2 2 3 2 1 1 1 2 pq1q2 pxq1yq2z p = 1, q1,q2 = 2,5 (9) 9455 [3],Type 3
3 0 2 1 3 0 4 3 0 0 2 2 3 pq1q2 pxq1yq2z p = 1, q1,q2 = 8 (9)

The cases without a reference are essentially unexplored.
However, for the types 1 and 2 in [2] also
more detailed results would be desirable.

 References: [0] Pierre Barrucand, Hugh C. Williams, and L. Baniuk, A computational technique for determining the class number of a pure cubic field, Math. Comp. 30 (1976),no. 134, 312 - 323 [1] Taira Honda, Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7 - 12 [2] Moulay Chrif Ismaïli, Sur la capitulation des 3-classes d'idéaux de la clôture normale d'un corps cubique pur, Thèse de doctorat, Université Laval, Québec, 1992 [3] Moulay Chrif Ismaïli and Rachid El Mesaoudi, Sur la divisibilité exacte par 3 du nombre de classes de certain corps cubiques purs, Ann. Sci. Math. Québec 25 (2001), no. 2, 153 - 177