A242873 -id:A242873 - cp's OEIS frontend

A242878 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 3, except for the cases mentioned in the COMMENTS.

9748, 15544, 16627, 17131, 18555, 21668, 22395, 22443, 23683, 24884, 27640, 28279, 31271, 34027, 34867, 35539, 37988, 39736, 42619, 42859, 43847, 45887, 48472, 48667, 50983, 51348, 53843, 54319, 58920, 60196, 60895

Offset: 1

Daniel Constantin Mayer, May 25 2014

CAVEAT: Up to 10^5, the length of the 3-tower is unknown for the following discriminants: 17131, 21668, 24884, 28279, 34027, 35539, 64952, 65203, 72591, 92660, 92827. The performance of the MAGMA script in section PROG would be much slower, if the class number of the first Hilbert 3-class field were computed. This would admit a criterion for the exclusion of the mentioned exceptional discriminants. Therefore, including the superfluous brushwood was the lesser of two evils.

			The case 9748 (n=1) was discussed very thoroughly by Scholz and Taussky in 1934. However, this is the famous case where they erroneously claimed that the 3-tower has exactly two stages. Brink and Gold had doubts about this claim but were unable to exclude it definitely in 1987. Bush and Mayer were the first who succeeded in disproving this claim rigorously in 2012.

J. R. Brink and R. Gold, Class field towers of imaginary quadratic fields, manuscripta math. 57 (1987), 425-450.
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, arXiv:1312.0251 [math.NT], J. Number Theory, accepted for publication, 2014
A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. Reine Angew. Math. 171 (1934), 19-41.

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 := 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; p := 0; 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 p := p+1; end if; end for; if (1 eq p) and ((0 eq e) or (1 eq e)) then d, ", "; end if; end if; end if; end for;

A242864 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 2.

Original entry on oeis.org

4027, 8751, 12131, 19187, 19651, 20276, 20568, 21224, 22711, 24340, 24904, 26139, 26760, 28031, 28759, 31639, 31999, 32968, 34088, 34507, 35367, 36807, 40299, 40692, 41015, 41583, 41671, 42423, 43192, 43307, 44004

Offset: 1

Daniel Constantin Mayer, May 24 2014

For all these discriminants, the metabelianization of the 3-tower group is one of the two Schur sigma-groups SmallGroup(243, 5) or SmallGroup(243, 7), whence it is clear that the tower must terminate at the second stage.
n = 1 is discussed very thoroughly by Scholz and Taussky.
These fields are characterized either by their 3-principalization types (transfer kernel types, TKTs) (2241), D.10, resp. (4224), D.5, or equivalently by their transfer target types (TTTs) [(3,3,3), (3,9)^3], resp. [(3,3,3)^2, (3,9)^2] (called IPADs by Boston, Bush, Hajir). The latter are used in the MAGMA PROG, which essentially constitutes the principalization algorithm via class group structure. - Daniel Constantin Mayer, Sep 23 2014

Laurent Bartholdi and Michael R. Bush, Maximal unramified 3-extensions of imaginary quadratic fields and SL_2Z_3, J. Number Theory, 124 (2007), 159-166.
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, The distribution of second p-class groups on coclass graphs, 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.
A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. Reine Angew. Math. 171 (1934), 19-41.

Cf. A242862, A242863 (supersequences), A247689, A247690 (subsequences), and A242873, A242878 (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) or (2 eq e)) then d,","; end if; end if; end if; end for;

A247694 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type H.4 (2122), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

Original entry on oeis.org

21668, 446788, 3843907, 52505588

Offset: 0

Daniel Constantin Mayer, Oct 12 2014

The 3-principalization type (transfer kernel type, TKT) H.4 (2122) is not a permutation, contains a transposition, and has no fixed point.
The nilpotency condition cl(G)=2n+6 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3^{n+2},3^{n+3}),(3,3,3),(3,9)^2].
The second 3-class group G is one of two vertices of depth 2 on the coclass tree with root SmallGroup(243,6) contained in the coclass graph G(3,2).
The length of the Hilbert 3-class field tower of all these fields is completely unknown. Therefore, these discriminants are among the foremost challenges of future research, similarly as those of A242873, A247688, A247697.
A247694 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.
The initial term a(0)=21668 has been recognized as a realization of TKT H.4 in the Dissertation of J. R. Brink(1984). However, Brink did not know that the TKT H.4 can also occur with second 3-class group G=SmallGroup(729,45) of nilpotency class cl(G)=4 and TTT [(3,3,3)^3,(3,9)]. Actually, D. C. Mayer (1991) was the first who proved that the integer 21668 is the smallest term of A247694 and does not belong to A242873.

			For a(0)=21668, we have the ground state of TKT H.4 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=6.
For a(1)=446788, we have the first excited state of TKT H.4 with TTT [(27,81),(3,3,3),(3,9)^2] and cl(G)=8.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).

J. R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, The Ohio State University, 1984.
D. C. Mayer, Principalization in complex S_3 fields, Congressus Numerantium 80 (1991), 73-87. (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, The University of Manitoba, Winnipeg, Manitoba, Canada, 1990.)

N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Preprint: arXiv:1111.4679v1 [math.NT], 2011, Math. Ann. (2013).
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
D. C. Mayer, The second p-class group of a number field
D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
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, Preprint: arXiv:1403.3839v1 [math.NT], 2014.
Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851 [math.NT], 2015.
Wikipedia, Artin transfer (group theory), Table 2

Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247695, A247696, A247697 (disjoint sequences).

A247697 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type G.16 (2134), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

Original entry on oeis.org

17131, 819743, 2244399, 30224744

Offset: 0

Daniel Constantin Mayer, Oct 12 2014

The 3-principalization type (transfer kernel type, TKT) G.16 (2134) is a permutation, contains a transposition, and has two fixed points.
The nilpotency condition cl(G)=2n+6 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3,9),(3^{n+2},3^{n+3}),(3,9)^2].
The second 3-class group G is one of two vertices of depth 2 on the coclass tree with root SmallGroup(243,8) contained in the coclass graph G(3,2).
The length of the Hilbert 3-class field tower of all these fields is completely unknown. Therefore, these discriminants are among the foremost challenges of future research, similarly as those of A242873, A247688, A247694.
A247697 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.

			For a(0)=17131, we have the ground state of TKT G.16 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=6.
For a(1)=819743, we have the first excited state of TKT G.16 with TTT [(3,9),(27,81),(3,9)^2] and cl(G)=8.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).

N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Preprint: arXiv:1111.4679v1 [math.NT], 2011; Math. Ann. (2013).
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
D. C. Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014.
D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
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, Preprint: arXiv:1403.3839v1 [math.NT], 2014.
Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851 [math.NT], 2015.
Wikipedia, Artin transfer (group theory), Table 2

Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247694, A247695, A247696 (disjoint sequences).

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.

Original entry on oeis.org

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;

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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A242878 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 3, except for the cases mentioned in the COMMENTS.

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A242864 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 2.

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A247694 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type H.4 (2122), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

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A247697 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type G.16 (2134), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

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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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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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