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A269318 Discriminants of real quadratic number fields with 3-class rank 2.

%I A269318 #20 Jan 27 2024 18:51:32
%S A269318 32009,42817,62501,72329,94636,103809,114889,130397,142097,151141,
%T A269318 152949,153949,172252,173944,184137,189237,206776,209765,213913,
%U A269318 214028,214712,219461,220217,250748,252977,255973,259653,265245,275881,282461,283673,298849,320785,321053,326945,333656,335229,341724,342664,358285,363397,371965,384369,390876
%N A269318 Discriminants of real quadratic number fields with 3-class rank 2.
%C A269318 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].
%H A269318 I. O. Angell, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0401701-6">A table of totally real cubic fields</a>, Math. Comp. 30 (1976), no. 133, 184-187.
%H A269318 M. R. Bush, <a href="http://home.wlu.edu/~bushm/Research/research.html">private communication</a>, 11 July 2015.
%H A269318 H. Cohen and J. Martinet, <a href="http://dx.doi.org/10.1090/S0025-5718-1987-0866103-4">Class groups of number fields: numerical heuristics</a>, Math. Comp. 48 (1987), no. 177, 123-137.
%H A269318 V. Ennola and R. Turunen, <a href="http://dx.doi.org/10.1090/S0025-5718-1985-0777281-8">On totally real cubic fields</a>, Math. Comp. 44 (1985), no. 170, 495-518.
%H A269318 P. Llorente and J. Quer, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0929555-8">On totally real cubic fields with discriminant D < 10^7</a>, Math. Comp. 50 (1988), no. 182, 581-594.
%H A269318 D. C. Mayer, <a href="http://dx.doi.org/10.1142/S1793042114500754">Quadratic p-ring spaces for counting dihedral fields</a>, Int. J. Number Theory 10 (2014), no. 8, 2205-2242.
%e A269318 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.
%o A269318 (Magma) SetClassGroupBounds("GRH"); p:=3;
%o A269318 for d:=0 to 10^6 do if ((d gt 1) and IsFundamental(d)) then
%o A269318 Q:=QuadraticField(d); O:=MaximalOrder(Q); C:=ClassGroup(O);
%o A269318 if (2 eq #pPrimaryInvariants(C,p)) then printf "%o,",d;
%o A269318 end if; end if; end for;
%Y A269318 Subsequence A269319.
%K A269318 nonn,easy
%O A269318 1,1
%A A269318 _Daniel Constantin Mayer_, Mar 06 2016