# 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, Quebec, 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 *

 Multiplicities of cubic discriminants (2002/02/05) Karim (02/01/29): I wrote that quick GP program I alluded to in my last email (implementing Hasse's criterion based on PARI's ray class group functions). I checked it on the fields you sent me: +89407866420 (43 subgroups of the correct conductor) +92303857800 (85 subgroups) +110903258700 (11 subgroups) -16008300 (11 subgroups) I (instantaneously) get multiplicity 5 in all cases [and so did my original program, whose components have finally all been merged !]. In all these 4 cases, only 5 subgroups are stable under s, and s acts non-trivially on Cl_f/H for all of them. Dan (02/01/31): I am very happy that you confirmed my old examples for multiplicity 5, which was unknown up to now (except to Pierre Barrucand). I found the pure cubic example -16008300 in Winnipeg City (1990) and the totally real ones (starting with +89407866420) in Bregenz (1992). I computed the latter by means of my regulator quotient criterion. Dan (02/02/03): Some further interesting cubic discriminants are: D = 5 * (2*3^2*17*19*23*31)^2 = 85920959629620 D = 17 * (5*3^2*13*19*23*29*41*43)^2 D = 28 * (5*3^2*11*19*23*31*37)^2 Karim (02/02/05): "Hasse lemma" program gets D = 5 * (2*3^2*17*19*23*31)^2 = 85920959629620 ...m = 11 D = 17 * (5*3^2*13*19*23*29*41*43)^2 ...m = 43 D = 28 * (5*3^2*11*19*23*31*37)^2 ...m = 21 respectively.