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A247691 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) whose second 3-class group is located on the sporadic part of the coclass graph G(3,2) outside of coclass trees. 0
3896, 4027, 6583, 8751, 12067, 12131, 19187, 19651, 20276, 20568, 21224, 22711, 23428, 24340, 24904, 25447, 26139, 26760, 27355, 27991, 28031, 28759, 31639, 31999, 32968, 34088, 34507, 35367, 36276, 36807, 37219, 37540, 39819, 40299, 40692, 41015, 41063, 41583, 41671, 42423, 43192 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These fields are characterized either by their 3-principalization types (transfer kernel types, TKTs) (2143), G.19, (2241), D.10, (4224), D.5, (4443), H.4, or equivalently by their transfer target types (TTTs) [(3,9)^4], [(3,3,3), (3,9)^3], [(3,3,3)^2, (3,9)^2], [(3,3,3)^3, (3,9)] (called IPADs by Boston, Bush, Hajir). The latter are used in the MAGMA PROG, which essentially constitutes the principalization algorithm via class group structure.

REFERENCES

D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.

D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (3-4) (2012), 467-495.

D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.

LINKS

Table of n, a(n) for n=1..41.

N. Boston, M. R. Bush, F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, arXiv:1111.4679 [math.NT], 2011, Math. Ann. (2013).

D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.

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; for j in [1..#sO] do CO := ClassGroup(sO[j]); if not (3 eq Valuation(#CO, 3)) then g := false; end if; end for; if (true eq g) then d, ", "; end if; end if; end if; end for;

CROSSREFS

Cf. A242862, A242863 (supersequences), A242864, A242873, A247688 (subsequences), and A242878 (disjoint sequence).

Sequence in context: A186555 A185862 A242863 * A242873 A135202 A204147

Adjacent sequences:  A247688 A247689 A247690 * A247692 A247693 A247694

KEYWORD

hard,nonn

AUTHOR

Daniel Constantin Mayer, Sep 27 2014

STATUS

approved

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Last modified January 17 07:18 EST 2018. Contains 297787 sequences.