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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<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;
CROSSREFS
Cf. A242862, A242863 (supersequences), A247689, A247690 (subsequences), and A242873, A242878 (disjoint sequences).
Sequence in context: A251098 A034229 A260246 * A247689 A258401 A258883
KEYWORD
hard,nonn
AUTHOR
STATUS
approved

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Last modified March 1 09:35 EST 2024. Contains 370432 sequences. (Running on oeis4.)