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A242864
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Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 2.
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6
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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PROG
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(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;
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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STATUS
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approved
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