A242863 -id:A242863 - cp's OEIS frontend

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

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;

A359871 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 5-class group (5,5).

Original entry on oeis.org

11199, 12451, 17944, 30263, 33531, 37363, 38047, 39947, 42871, 53079, 54211, 58424, 61556, 62632, 63411, 64103, 65784, 66328, 67031, 67063, 67128, 69811, 72084, 74051, 75688, 83767, 84271, 85099, 85279, 87971, 89751, 90795, 90868, 92263, 98591, 99031, 99743

Offset: 1

Daniel Constantin Mayer, Jan 16 2023

The maximal unramified pro-5-extension, that is, the Hilbert 5-class field tower, of these imaginary quadratic fields must have a Schur sigma-group as its Galois group. The tower has an unbounded number of stages at least equal to two, and may even be infinite.

			On page 22 of their 1982 paper, Franz-Peter Heider and Bodo Schmithals gave the smallest prime discriminant -12451 and determined two of the six capitulation kernels in unramified cyclic quintic extensions. On 03 November 2011, Daniel C. Mayer determined the abelian type invariants, and thus indirectly the coarse capitulation type, of these six extensions for all 37 discriminants in the range between -11199 and -99743, with computational aid by Claus Fieker. In particular, -89751 was the minimal occurrence of the identity capitulation (see A359291). In his 2012 Ph.D. thesis, Tobias Bembom independently recomputed the capitulation in this range, without being able to detect the identity capitulation for -89751. It must be pointed out that in his table on pages 129 and 130, the minimal discriminant -11199=-3*3733 is missing, whereas the discriminant -81287 is superfluous and must be cancelled, since its 5-class group is non-elementary bicyclic of type (25,5).

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 (2013), no. 2, 401-456. (Sec. 3.5.2, p. 448)

T. Bembom, The capitulation problem in class field theory, Dissertation, Univ. Göttingen, 2012. (Sec. 6.3, p. 128)
D. C. Mayer, Pentadic quantum class groups
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014. (Sec. 3.5.2-3.5.3, pp. 448-450)

Cf. A359291 (subsequence), A242863 (3,3), A359872 (7,7).

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 ([5,5] eq pPrimaryInvariants(C,5)) then d, ", "; end if; end if; end for;

A359872 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 7-class group (7,7).

Original entry on oeis.org

63499, 118843, 124043, 149519, 159592, 170679, 183619, 185723, 220503, 226691, 227387, 227860, 236931, 240347, 240655, 247252, 260111, 268739, 272179, 275636, 294935, 299627, 301211, 308531, 318547, 346883, 361595, 366295, 373655, 465719, 489576, 491767, 501576, 506551, 511988, 518879, 528243, 546792, 553791

Offset: 1

Daniel Constantin Mayer, Jan 16 2023

The maximal unramified pro-7-extension, that is, the Hilbert 7-class field tower, of these imaginary quadratic fields must have a Schur sigma-group as its Galois group. The tower has an unbounded number of stages at least equal to two, and may even be infinite.

			On 06 January 2012, Daniel C. Mayer determined the abelian type invariants (ATI), and thus indirectly the coarse capitulation type, of the eight unramified cyclic septic extensions for all 70 discriminants in the range between -63499 and -751288. On page 133 of his 2012 Ph.D. thesis, Tobias Bembom independently recomputed the capitulation for the two discriminants -63499 and -159592. In the time between 09 and 16 August 2014, Daniel C. Mayer computed the fine capitulation type of all 94 discriminants in the range -63499 and -991720 without any hit of the identity capitulation. Since the fine capitulation requires much more CPU time than the ATI, Mayer conducted an extensive search for the identity capitulation, identified by eight ATI of the shape (7,7,7), in the range from 10^6 to 6578723, with an eventual successful hit of the identity capitulation for -5073691 (the 555th term of A359872) on 26 October 2019 (see A359296).

Daniel C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401-456. (Sec. 3.5.2, p. 448)

Tobias Bembom, The capitulation problem in class field theory, Dissertation, Univ. Göttingen, 2012. (Sec. 6.3, p. 128)
Daniel C. Mayer, Heptadic quantum class groups
Daniel C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014. (Sec. 3.5.4, pp. 450-451)

Cf. A359296 (subsequence), A242863 (3,3), A359871 (5,5).

for d := 2 to 10^6 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 ([7,7] eq pPrimaryInvariants(C,7)) then d, ", "; end if; end if; end for;

A244575 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3,3), thus having an infinite class tower.

Original entry on oeis.org

4447704, 4472360, 4818916, 4897363, 5067967, 5769988, 7060148, 8180671, 8721735, 8819519, 8992363, 9379703, 9487991, 9778603

Offset: 1

Daniel Constantin Mayer, Jun 30 2014

I do not know who actually discovered a(1)=4447704. It is mentioned neither in Diaz y Diaz (1973) nor in Buell (1976). Maybe it can be found in Shanks (1976). Magma required 18 hours CPU time for the first 14 terms.

Meanwhile, it came to my attention that a(1)=4447704 and all the other terms below 10^7 are given in Appendice 1, pp. 66-77, of the Thesis of Diaz y Diaz (1978). a(1) is not contained in Shanks (1976). - Daniel Constantin Mayer, Sep 28 2014.

			a(1)=4447704 is the minimal absolute discriminant with elementary abelian 3-class group of type (3,3,3), whereas the smaller A244574(1)=3321607 has non-elementary (9,3,3).

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.
F. 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.
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 [math.NT], 2015.
D. Shanks, Class groups of the quadratic fields found by Diaz y Diaz, Math. Comp. 30 (1976), 173-178.

Cf. A242863, A244574 (a supersequence).

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 ([3,3,3] 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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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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A359871 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 5-class group (5,5).

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A359872 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 7-class group (7,7).

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A244575 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3,3), thus having an 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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