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%I #20 Sep 08 2022 08:46:15
%S 32009,42817,62501,72329,94636,103809,114889,130397,142097,151141,
%T 152949,153949,172252,173944,184137,189237,206776,209765,213913,
%U 214028,214712,219461,220217,250748,252977,259653,265245,275881,283673,298849,320785,321053,326945,333656,335229,341724,342664,358285,363397,371965,390876
%N Discriminants of real quadratic fields with 3-class group of type (3,3)
%C As explained in the comments in A269318, the terms of A269319 are discriminants of quadratic fields K which correspond to certain 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), 58 in [Ennola, Turunen] (up to 5*10^5), and 2576 in [Llorente, Quer] (up to 10^7). It should be pointed out that, whereas [Angell] does not contain other quartets than the 5 corresponding to type (3,3), there occur 3 further quartets associated with type (9,3) in [Ennola, Turunen], namely 255973, 282461, 384369. In [Llorente, Quer], we have 271 additional quartets of type (9,3), 20 of type (27,3), 1 of type (81,3), and 2 of type (9,9). The splitting 2879-9=2870=2576+271+20+1+2 was computed in [Mayer, 2010] and is not contained in [Llorente, Quer]. The number 2576 was published in [Mayer, 2012] and is not mentioned in [Llorente, Quer]. The most recent and most extensive information is due to [Bush], who showed that there are 415698 quartets associated with type (3,3) up to the bound 10^9.
%H 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 M. R. Bush, <a href="http://home.wlu.edu/~bushm/Research/research.html">private communication</a>, on 11 July 2015.
%H 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 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 D. C. Mayer, <a href="http://www.algebra.at/SciRes2010TopDown.htm">Top down capitulation algorithm</a>, Scientific Research 2010.
%H D. C. Mayer, <a href="http://dx.doi.org/10.1142/S179304211250025X">The second p-class group of a number field</a>, Int. J. Number Theory 8 (2012), no. 2, 471-505.
%e The execution of the MAGMA program requires the supersequence A269318 as its input list, and yields the 149 leading terms of A269319 up to 10^6, sifting out 12 terms with associated 3-class group of type (9,3).
%o (Magma) SetClassGroupBounds("GRH"); p:=3; dList:=A269318; for d in dList do
%o ZX<X>:=PolynomialRing(Integers()); K:=NumberField(X^2-d); O:=MaximalOrder(K); C:=ClassGroup(O); if ([p,p] eq pPrimaryInvariants(C,p)) then printf "%o,",d; end if; end for;
%Y Subsequence of A269318, contains disjoint subsequences A269320,...,A269323
%K nonn,easy
%O 1,1
%A _Daniel Constantin Mayer_, Mar 06 2016
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