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A247689 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (2241). 2
4027, 8751, 19651, 21224, 22711, 24904, 26139, 28031, 28759, 34088, 36807, 40299, 40692, 41015, 42423, 43192, 44004, 45835, 46587, 48052, 49128, 49812, 50739, 50855, 51995, 55247, 55271, 55623, 70244, 72435, 77144, 78708, 81867, 85199, 87503, 87727, 88447, 91471, 91860, 92712, 94420, 95155, 97555, 98795, 99707, 99939 (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) (2241), D.10, or equivalently by their transfer target type (TTT) [(3,3,3), (3,9)^3] (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 (2241) has a single fixed point and is not a permutation.
For all these discriminants, the 3-tower group is the metabelian Schur sigma-group SmallGroup(243, 5) and the Hilbert 3-class field tower terminates at the second stage.
4027 is discussed very thoroughly by Scholz and Taussky.
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, Math. Ann. (2013).
D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
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) then d, ", "; end if; end if; end if; end for;
CROSSREFS
Cf. A242862, A242863, A242864 (supersequences), and A247690, A242873 (disjoint sequences).
Sequence in context: A034229 A260246 A242864 * A258401 A258883 A352752
KEYWORD
hard,nonn
AUTHOR
STATUS
approved

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Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)