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A242864 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 2. 6

%I #31 Sep 08 2022 08:46:08

%S 4027,8751,12131,19187,19651,20276,20568,21224,22711,24340,24904,

%T 26139,26760,28031,28759,31639,31999,32968,34088,34507,35367,36807,

%U 40299,40692,41015,41583,41671,42423,43192,43307,44004

%N Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 2.

%C For all these discriminants, the metabelianization of the 3-tower group is one of the two Schur sigma-groups SmallGroup(243, 5) or SmallGroup(243, 7), whence it is clear that the tower must terminate at the second stage.

%C n = 1 is discussed very thoroughly by Scholz and Taussky.

%C These fields are characterized either by their 3-principalization types (transfer kernel types, TKTs) (2241), D.10, resp. (4224), D.5, or equivalently by their transfer target types (TTTs) [(3,3,3), (3,9)^3], resp. [(3,3,3)^2, (3,9)^2] (called IPADs by Boston, Bush, Hajir). The latter are used in the MAGMA PROG, which essentially constitutes the principalization algorithm via class group structure. Daniel Constantin Mayer, September 23 2014

%H Laurent Bartholdi and Michael R. Bush, <a href="http://dx.doi.org/10.1016/j.jnt.2006.08.008">Maximal unramified 3-extensions of imaginary quadratic fields and SL_2Z_3</a>, J. Number Theory, 124 (2007), 159-166.

%H N. Boston, M. R. Bush, F. Hajir, <a href="http://arxiv.org/abs/1111.4679">Heuristics for p-class towers of imaginary quadratic fields</a>, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.

%H D. C. Mayer, <a href="http://dx.doi.org/10.5802/jtnb.842">The distribution of second p-class groups on coclass graphs</a>, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.

%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.

%H A. Scholz and O. Taussky, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002172852&amp;IDDOC=253437">Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper</a>, J. Reine Angew. Math. 171 (1934), 19-41.

%o (Magma)

%o for d := 2 to 10^5 do a := false; if (3 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (1 eq r mod 4)) then a := true; end if; end if; if (true eq a) then K := QuadraticField(-d); C,mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then E := AbelianExtension(mC); sS := Subgroups(C: Quot := [3]); sA := [AbelianExtension(Inverse(mQ)*mC) where Q,mQ := quo<C|x`subgroup>: x in sS]; sN := [NumberField(x): x in sA]; sF := [AbsoluteField(x): x in sN]; sM := [MaximalOrder(x): x in sF]; sM := [OptimizedRepresentation(x): x in sF]; sA := [NumberField(DefiningPolynomial(x)): x in sM]; sO := [Simplify(LLL(MaximalOrder(x))): x in sA]; delete sA,sN,sF,sM; g := true; e := 0; for j in [1..#sO] do CO := ClassGroup(sO[j]); if (3 eq Valuation(#CO,3)) then if ([3,3,3] eq pPrimaryInvariants(CO,3)) then e := e+1; end if; else g := false; end if; end for; if (true eq g) and ((1 eq e) or (2 eq e)) then d,","; end if; end if; end if; end for;

%Y Cf. A242862, A242863 (supersequences), A247689, A247690 (subsequences), and A242873, A242878 (disjoint sequences).

%K hard,nonn

%O 1,1

%A _Daniel Constantin Mayer_, May 24 2014

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)