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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
 4027, 8751, 12131, 19187, 19651, 20276, 20568, 21224, 22711, 24340, 24904, 26139, 26760, 28031, 28759, 31639, 31999, 32968, 34088, 34507, 35367, 36807, 40299, 40692, 41015, 41583, 41671, 42423, 43192, 43307, 44004 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. n = 1 is discussed very thoroughly by Scholz and Taussky. 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 LINKS Laurent Bartholdi and Michael R. Bush, Maximal unramified 3-extensions of imaginary quadratic fields and SL_2Z_3, J. Number Theory, 124 (2007), 159-166. N. Boston, M. R. Bush, F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011. D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456. D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014. A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. Reine Angew. Math. 171 (1934), 19-41. PROG (MAGMA) 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: 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; CROSSREFS Cf. A242862, A242863 (supersequences), A247689, A247690 (subsequences), and A242873, A242878 (disjoint sequences). Sequence in context: A251098 A034229 A260246 * A247689 A258401 A258883 Adjacent sequences:  A242861 A242862 A242863 * A242865 A242866 A242867 KEYWORD hard,nonn AUTHOR Daniel Constantin Mayer, May 24 2014 STATUS approved

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Last modified October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)