A242862 -id:A242862 - cp's OEIS frontend

A247688 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3), 3-principalization type (2143), IPAD [(3,9)^4], and Hilbert 3-class field tower of unknown length at least 3.

12067, 49924, 54195, 60099, 83395, 86551, 91643, 93067, 96551

Offset: 1

Daniel Constantin Mayer, Sep 22 2014

These fields are characterized either by their 3-principalization type (transfer kernel type, TKT) (2143), G.19, or equivalently by their transfer target type (TTT) [(3,9)^4] (called IPAD by Boston, Bush, Hajir). The latter is used in the MAGMA PROG. The TKT (2143) is a permutation composed of two disjoint transpositions without fixed point.
For all these discriminants, the metabelianization of the 3-tower group is the unbalanced group SmallGroup(729, 57), whence it is completely open whether the tower must terminate at a finite stage or not. Consequently, these discriminants are among the foremost challenges of future research.
12067 has been discovered by Heider and Schmithals.

			Already the smallest term 12067 resists all attempts to determine the length of its Hilbert 3-class field tower.

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.

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.

Cf. A242862, A242863 (supersequences), and A242864, A242873 (disjoint sequences).

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: 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 (0 eq e) then d, ", "; end if; end if; end if; end for;

A247689 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (2241).

Original entry on oeis.org

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

Offset: 1

Daniel Constantin Mayer, Sep 23 2014

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.

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.

Cf. A242862, A242863, A242864 (supersequences), and A247690, A242873 (disjoint sequences).

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: 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;

A247690 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (4224).

Original entry on oeis.org

12131, 19187, 20276, 20568, 24340, 26760, 31639, 31999, 32968, 34507, 35367, 41583, 41671, 43307, 57079, 64196, 73731, 85796, 87720, 93823, 95691

Offset: 1

Daniel Constantin Mayer, Sep 23 2014

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.

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.

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.

Cf. A242862, A242863, A242864 (supersequences), and A247689, A242873 (disjoint sequences).

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: 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;

A244574 Absolute discriminants of complex quadratic fields with 3-class rank 3 and thus with infinite class tower.

Original entry on oeis.org

3321607, 3640387, 4019207, 4447704, 4472360, 4818916, 4897363, 5048347, 5067967, 5153431, 5288968, 5769988, 6562327, 7016747, 7060148, 7503391, 7546164, 8124503, 8180671, 8721735, 8819519, 8992363, 9379703, 9487991, 9778603

Offset: 1

Daniel Constantin Mayer, Jun 30 2014

Diaz y Diaz discovered a(1), a(2) and three other terms in 1973. However, Buell was the first who proved minimality of a(1). According to Koch and Venkov, 3-class rank 3 ensures an infinite Hilbert (3-)class field tower.
The first 25 terms were computed with MAGMA over 18 hours of CPU time.
With exception of a(16)=7503391, all terms below 10^7 and lots of further terms below 10^8 are given in Appendice 1, pp. 66-77, of the Thesis of F. Diaz y Diaz (1978). - Daniel Constantin Mayer, Sep 27 2014

			3-class group of type (9,3,3) for a(1)=3321607, and of type (3,3,3) for a(4)=4447704. Unique 3-class group of type (27,3,3) for a(10)=5153431.

F. Diaz y Diaz, Sur le 3-rang des corps quadratiques, Publ. math. d'Orsay, No. 78-11, Univ. Paris-Sud (1978).

D. A. Buell, Class groups of quadratic fields, Math. Comp. 30 (1976), no. 135, 610-623.
Francisco Diaz y Diaz, Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supérieur à 1, Séminaire Delange-Pisot-Poitou, 1973/74, no. G15.
H. Koch, B. B. Venkov, Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24-25 (1975), 57-67.
D. C. Mayer, Complex quadratic fields of type (3, 3, 3), 2014.
Daniel C. Mayer, Index-p abelianization data of p-class tower groups, arXiv preprint arXiv:1502.03388, 2015

Cf. A242862, A244575 (a subsequence).

for d := 1 to 10^7 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 eq #pPrimaryInvariants(C,3)) then d,","; end if; end if; end for;

A380102 Minimal absolute discriminants |d| of imaginary quadratic number fields K = Q(sqrt(d)), d < 0, with elementary bicyclic 3-class group Cl_3(K)=(3,3) and second 3-class group M=Gal(F_3^2(K)/K) of assigned even coclass cc(M)=2,4,6,8,...

Original entry on oeis.org

3896, 27156, 423640, 99888340

Offset: 1

Daniel Constantin Mayer, Jan 12 2025

The coclass cc(M) for the field K with discriminant d = -a(n) is 2*n, and for each field K with absolute discriminant |d| < a(n), the coclass cc(M) is less than 2*n.

			The coclass cannot be odd for imaginary quadratic fields. We have cc(M)=2 for d=-3896, cc(M)=4 for d=-27156, cc(M)=6 for d=-423640, cc(M)=8 for d=-99888340.

D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2013), 401-456.
Daniel Constantin Mayer, Magma program "ImagCoClass.m" with endless loop

Cf. A242862, A242863 (supersequences). Analog of A379524 for real quadratic fields.

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According to Theorem 3.12 on page 435 of "The distribution of second p-class groups on coclass graphs", the coclass of the group M is given by cc(M)+1=log_3(h_3(L_2)), where h_3(L_2) is the second largest 3-class number among the four unramified cyclic cubic extensions L_1,..,L_4 of the quadratic field K, and log_3 denotes the logarithm with respect to the basis 3.

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.

Original entry on oeis.org

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

Offset: 1

Daniel Constantin Mayer, Sep 27 2014

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.

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 second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.

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

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: 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;

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A247688 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3), 3-principalization type (2143), IPAD [(3,9)^4], and Hilbert 3-class field tower of unknown length at least 3.

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A247689 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (2241).

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A247690 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (4224).

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A244574 Absolute discriminants of complex quadratic fields with 3-class rank 3 and thus with infinite class tower.

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A380102 Minimal absolute discriminants |d| of imaginary quadratic number fields K = Q(sqrt(d)), d < 0, with elementary bicyclic 3-class group Cl_3(K)=(3,3) and second 3-class group M=Gal(F_3^2(K)/K) of assigned even coclass cc(M)=2,4,6,8,...

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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.

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