A013658 -id:A013658 - cp's OEIS frontend

A192322 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 4-group, C2 x C2.

84, 120, 132, 168, 195, 228, 280, 312, 340, 372, 408, 435, 483, 520, 532, 555, 595, 627, 708, 715, 760, 795, 1012, 1435

Offset: 1

David Terr, Jun 28 2011

Added keyword "full" - This sequence is a subsequence of A013658, whose last term is 1555. I have verified that the terms above are complete and correct. - Rick L. Shepherd, May 06 2013

Alexandre Gélin, Everett W. Howe, and Christophe Ritzenthaler, Principally Polarized Squares of Elliptic Curves with Field of Moduli Equal To Q, arXiv:1806.03826 [math.NT], 2018 (see table 1 page 4).
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Subsequence of A013658.

ok(n)={isfundamental(-n) && [2, 2] == quadclassunit(-n).cyc} \\ Andrew Howroyd, Jul 20 2018

A046003 Discriminants of imaginary quadratic fields with class number 6 (negated).

Original entry on oeis.org

87, 104, 116, 152, 212, 244, 247, 339, 411, 424, 436, 451, 472, 515, 628, 707, 771, 808, 835, 843, 856, 1048, 1059, 1099, 1108, 1147, 1192, 1203, 1219, 1267, 1315, 1347, 1363, 1432, 1563, 1588, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563, 2787, 2923, 3235, 3427, 3523, 3763

Offset: 1

Eric W. Weisstein

Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998) 295-330.
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Cf. A191410.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[3800], NumberFieldClassNumber[Sqrt[-#]] == 6 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 6};
for(n=1, 4000, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

[n for n in (1..4000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==6] # G. C. Greubel, Mar 01 2019

More terms from Seiichi Manyama, Jun 03 2018

A046007 Discriminants of imaginary quadratic fields with class number 10 (negated).

Original entry on oeis.org

119, 143, 159, 296, 303, 319, 344, 415, 488, 611, 635, 664, 699, 724, 779, 788, 803, 851, 872, 916, 923, 1115, 1268, 1384, 1492, 1576, 1643, 1684, 1688, 1707, 1779, 1819, 1835, 1891, 1923, 2152, 2164, 2363, 2452, 2643, 2776, 2836, 2899, 3028

Offset: 1

Eric W. Weisstein

87 discriminants in this sequence (almost certainly but not proved).

Andrew Howroyd, Table of n, a(n) for n = 1..87
Steven Arno, M. L. Robinson and Ferrel S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arithm. 83.4 (1998), 295-330
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Cf. A191410.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[14000], NumberFieldClassNumber[Sqrt[-#]] == 10 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 10} \\ Andrew Howroyd, Jul 24 2018

[n for n in (1..3500) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==10] # G. C. Greubel, Mar 01 2019

A046009 Discriminants of imaginary quadratic fields with class number 12 (negated).

Original entry on oeis.org

231, 255, 327, 356, 440, 516, 543, 655, 680, 687, 696, 728, 731, 744, 755, 804, 888, 932, 948, 964, 984, 996, 1011, 1067, 1096, 1144, 1208, 1235, 1236, 1255, 1272, 1336, 1355, 1371, 1419, 1464, 1480, 1491, 1515, 1547, 1572, 1668, 1720, 1732

Offset: 1

Eric W. Weisstein

206 discriminants in this sequence (almost certainly but not proved).

Andrew Howroyd, Table of n, a(n) for n = 1..206
Steven Arno, M. L. Robinson and Ferrel S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arithm. 83.4 (1998), 295-330
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Cf. A191410.

Reap[ For[n = 1, n < 2000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 12, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 12} \\ Andrew Howroyd, Jul 24 2018

[n for n in (1..3000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==12] # G. C. Greubel, Mar 01 2019

A046011 Discriminants of imaginary quadratic fields with class number 14 (negated).

Original entry on oeis.org

215, 287, 391, 404, 447, 511, 535, 536, 596, 692, 703, 807, 899, 1112, 1211, 1396, 1403, 1527, 1816, 1851, 1883, 2008, 2123, 2147, 2171, 2335, 2427, 2507, 2536, 2571, 2612, 2779, 2931, 2932, 3112, 3227, 3352, 3579, 3707, 3715, 3867, 3988

Offset: 1

Eric W. Weisstein

There are 95 discriminants in this sequence (almost certainly but not proved).

Andrew Howroyd, Table of n, a(n) for n = 1..95
Steven Arno, M. L. Robinson and Ferrel S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arithm. 83.4 (1998), 295-330
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Reap[ For[n = 1, n < 4000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 14, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 14} \\ Andrew Howroyd, Jul 24 2018

A046013 Discriminants of imaginary quadratic fields with class number 16 (negated).

Original entry on oeis.org

399, 407, 471, 559, 584, 644, 663, 740, 799, 884, 895, 903, 943, 1015, 1016, 1023, 1028, 1047, 1139, 1140, 1159, 1220, 1379, 1412, 1416, 1508, 1560, 1595, 1608, 1624, 1636, 1640, 1716, 1860, 1876, 1924, 1983, 2004, 2019, 2040, 2056, 2072

Offset: 1

Eric W. Weisstein

322 discriminants in this sequence (almost certainly but not proved).

Andrew Howroyd, Table of n, a(n) for n = 1..322
Steven Arno, M. L. Robinson and Ferrel S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arithm. 83.4 (1998), 295-330
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Victor Y. Wang, On Hilbert 2-class fields and 2-towers of imaginary quadratic number fields, arXiv preprint arXiv:1508.06552, 2015
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Reap[ For[n = 1, n < 3000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 16, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

ok(n)={isfundamental(-n) && qfbclassno(-n) == 16} \\ Andrew Howroyd, Jul 24 2018

A046015 Discriminants of imaginary quadratic fields with class number 18 (negated).

Original entry on oeis.org

335, 519, 527, 679, 1135, 1172, 1207, 1383, 1448, 1687, 1691, 1927, 2047, 2051, 2167, 2228, 2291, 2315, 2344, 2644, 2747, 2859, 3035, 3107, 3543, 3544, 3651, 3688, 4072, 4299, 4307, 4568, 4819, 4883, 5224, 5315, 5464, 5492, 5539, 5899

Offset: 1

Eric W. Weisstein

The class group of Q[sqrt(-d)] is isomorphic to C_3 X C_6 for d = 9748, 12067, 16627, 17131, 19651, 22443, 23683, 34027, 34507. For all other known d in this sequence, the class group of Q[sqrt(-d)] is isomorphic to C_18. - Jianing Song, Dec 01 2019

Jianing Song, Table of n, a(n) for n = 1..150
Steven Arno, M. L. Robinson and Ferrel S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arithm. 83.4 (1998), 295-330
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Reap[ For[n = 1, n < 6000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 18, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A329182 Negative discriminants with form class group isomorphic to C_2 X C_2 (negated).

Original entry on oeis.org

84, 96, 120, 132, 160, 168, 180, 192, 195, 228, 240, 280, 288, 312, 315, 340, 352, 372, 408, 435, 448, 483, 520, 532, 555, 595, 627, 708, 715, 760, 795, 928, 1012, 1435

Offset: 1

Jianing Song, Dec 05 2019

This sequence is finite and this is the full list.

Equivalently, negative discriminants of orders whose class group is isomorphic to C_2 X C_2 (negated). - Jianing Song, May 17 2021

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), A322710 (isomorphic to C_2), A328825 (isomorphic to C_3), this sequence (isomorphic to C_2 X C_2), A330219 (isomorphic to C_4).

The fundamental terms are listed in A192322. Cf. also A013658.

isA329182(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[2,2] \\ Jianing Song, May 17 2021

A330219 Negative discriminants with form class group isomorphic to C_4 (negated).

Original entry on oeis.org

39, 55, 56, 63, 68, 80, 128, 136, 144, 155, 156, 171, 184, 196, 203, 208, 219, 220, 252, 256, 259, 275, 291, 292, 323, 328, 355, 363, 387, 388, 400, 475, 507, 568, 592, 603, 667, 723, 763, 772, 955, 1003, 1027, 1227, 1243, 1387, 1411, 1467, 1507, 1555

Offset: 1

Jianing Song, Dec 05 2019

It seems that this is the full list.

Equivalently, negative discriminants of orders whose class group is isomorphic to C_4 (negated). - Jianing Song, May 17 2021

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), A322710 (isomorphic to C_2), A328825 (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), this sequence (isomorphic to C_4).

Subsequence of A133676 and A317987. Cf. also A013658.

isA330219(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[4] \\ Jianing Song, May 17 2021

A123563 Discriminants of imaginary quadratic fields with class number 20 (negated).

Original entry on oeis.org

455, 615, 776, 824, 836, 920, 1064, 1124, 1160, 1263, 1284, 1460, 1495, 1524, 1544, 1592, 1604, 1652, 1695, 1739, 1748, 1796, 1880, 1887, 1896, 1928, 1940, 1956, 2136, 2247, 2360, 2404, 2407, 2483, 2487, 2532, 2552

Offset: 1

Eric W. Weisstein, Nov 19 2006

A finite sequence with exactly 350 terms.

Eric Weisstein's World of Mathematics, Class Number.

Cf. A006203, A013658, A014602, A014603, A046002, ..., A046016, A046018, A046020.

Reap[ For[n = 1, n < 3000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 20, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

cp's OEIS Frontend

A192322 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 4-group, C2 x C2.

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A046003 Discriminants of imaginary quadratic fields with class number 6 (negated).

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A046007 Discriminants of imaginary quadratic fields with class number 10 (negated).

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A046009 Discriminants of imaginary quadratic fields with class number 12 (negated).

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A046011 Discriminants of imaginary quadratic fields with class number 14 (negated).

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A046013 Discriminants of imaginary quadratic fields with class number 16 (negated).

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A046015 Discriminants of imaginary quadratic fields with class number 18 (negated).

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A329182 Negative discriminants with form class group isomorphic to C_2 X C_2 (negated).

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