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A242863
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Absolute discriminants of complex quadratic fields with 3-class group of elementary abelian type (3,3) of rank 2.
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16
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3896, 4027, 6583, 8751, 9748, 12067, 12131, 15544, 16627, 17131, 18555, 19187, 19651, 20276, 20568, 21224, 21668, 22395, 22443, 22711, 23428, 23683, 24340, 24884, 24904, 25447, 26139, 26760, 27156, 27355, 27640
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OFFSET
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1,1
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COMMENTS
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This is the best studied subsequence of A242862. For all these discriminants, the metabelianization of the 3-tower group is known. For two extensive subsequences the 3-class tower has exact length 2, resp. 3.
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REFERENCES
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F.-P. Heider, B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25.
B. Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Univ. zu Köln, 1989.
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LINKS
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EXAMPLE
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The exact length of the 3-class field tower is 2 for n=2,4,7, and 3 for n=5,8,9.
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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 := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;
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CROSSREFS
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Cf. A242862 (supersequence with arbitrary 3-class rank 2).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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