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A247690 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (4224). 2
12131, 19187, 20276, 20568, 24340, 26760, 31639, 31999, 32968, 34507, 35367, 41583, 41671, 43307, 57079, 64196, 73731, 85796, 87720, 93823, 95691 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These fields are characterized either by their 3-principalization type (transfer kernel type, TKT) (4224), D.5, or equivalently by their transfer target type (TTT) [(3,3,3)^2, (3,9)^2] (called IPAD by Boston, Bush, Hajir). The latter is used in the MAGMA PROG, which essentially constitutes the principalization algorithm via class group structure. The TKT (4224) has two fixed points and is not a permutation.
For all these discriminants, the 3-tower group is the metabelian Schur sigma-group SmallGroup(243, 7) and the Hilbert 3-class field tower terminates at the second stage.
12131 has been discovered by Heider and Schmithals.
REFERENCES
F.-P. Heider, B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25.
D. C. Mayer, "The distribution of second p-class groups on coclass graphs", J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
LINKS
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, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.
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 (2 eq e) then d, ", "; end if; end if; end if; end for;
CROSSREFS
Cf. A242862, A242863, A242864 (supersequences), and A247689, A242873 (disjoint sequences).
Sequence in context: A247403 A262635 A186564 * A126847 A018235 A250837
KEYWORD
hard,more,nonn
AUTHOR
STATUS
approved

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Last modified June 19 03:10 EDT 2024. Contains 373492 sequences. (Running on oeis4.)