OFFSET
1,1
COMMENTS
The number of unramified cyclic extensions N|K of relative degree p of a quadratic field K with p-class rank r (p an odd prime) is given by the multiplicity formula m = (p^r-1)/(p-1) [Mayer, Theorem 3.1]. Here, we have p=3, r=2, and thus m=4. Consequently, the terms of A269318 characterize all quartets (L_1, ..., L_4) of pairwise non-isomorphic non-Galois cubic fields sharing a common fundamental discriminant d(L_i) = d(K). There occur 5 of these quartets in [Angell] (up to 10^5), 61 in [Ennola, Turunen] (up to 5*10^5), and 2870 in [Llorente, Quer] (up to 10^7). The number 2879 in the first and third line below Table 4 [Llorente, Quer] is erroneous, since the 9 quartets in Table 6 [Llorente, Quer] are ramified and satisfy d(L_i) = f^2*d(K) with various conductors f > 1. (We point out misprints in the caption and in the header of Table 6 [Llorente, Quer], where our Fuehrer f is denoted by T and should correctly be given by 3^m*T_0.) The most recent and most extensive computation is due to [Bush]. He found 481756 unramified quartets up to 10^9, which are obviously very sparse with absolute density ~0.05%. The density ~0.16% with respect to the asymptotic number (3/Pi^2)*10^9 ~ 303963551 of all positive fundamental discriminants is slightly bigger. Compare the Cohen-Lenstra heuristics [Cohen, Martinet].
LINKS
I. O. Angell, A table of totally real cubic fields, Math. Comp. 30 (1976), no. 133, 184-187.
M. R. Bush, private communication, 11 July 2015.
H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), no. 177, 123-137.
V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495-518.
P. Llorente and J. Quer, On totally real cubic fields with discriminant D < 10^7, Math. Comp. 50 (1988), no. 182, 581-594.
D. C. Mayer, Quadratic p-ring spaces for counting dihedral fields, Int. J. Number Theory 10 (2014), no. 8, 2205-2242.
EXAMPLE
The execution of the Magma program yields the 161 leading terms of A269318 up to 10^6 and requires 9200 seconds on a single thread of an Intel i7 4-core processor with clock frequency 4GHz. The computation is slow because 303957 discriminants have to be checked for the structure of their associated 3-class groups. Among the 161 3-class groups of 3-rank 2, there are 149 of type (3,3) and 12 of type (9,3). Parallelization (for instance, 4 threads processing ranges of length 250000) would reduce the CPU time.
PROG
(Magma) SetClassGroupBounds("GRH"); p:=3;
for d:=0 to 10^6 do if ((d gt 1) and IsFundamental(d)) then
Q:=QuadraticField(d); O:=MaximalOrder(Q); C:=ClassGroup(O);
if (2 eq #pPrimaryInvariants(C, p)) then printf "%o, ", d;
end if; end if; end for;
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Daniel Constantin Mayer, Mar 06 2016
STATUS
approved