A192322 -id:A192322 - cp's OEIS frontend

A013658 Discriminants of imaginary quadratic fields with class number 4 (negated).

39, 55, 56, 68, 84, 120, 132, 136, 155, 168, 184, 195, 203, 219, 228, 259, 280, 291, 292, 312, 323, 328, 340, 355, 372, 388, 408, 435, 483, 520, 532, 555, 568, 595, 627, 667, 708, 715, 723, 760, 763, 772, 795, 955, 1003, 1012, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555

Offset: 1

Eric Rains (rains(AT)caltech.edu)

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 229.

Giovanni Resta, Table of n, a(n) for n = 1..54 (full sequence, from Weisstein's World of Mathematics)
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Class Number
Sung Sik Woo, Cubic formula and cubic curves, Commun. Korean Math. Soc. 28 (2013), No. 2, pp. 209-224.
Index entries for sequences related to quadratic fields

Cf. A014603, A046005, A192322.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[1250], NumberFieldClassNumber[Sqrt[-#]] == 4 &]] (* Jean-François Alcover, Jun 27 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 4} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..2000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==4] # G. C. Greubel, Mar 01 2019

a(50)-a(54) added by Andrew Howroyd, Jul 20 2018

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

A305416 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 8-group, C_2 x C_2 x C_2.

Original entry on oeis.org

420, 660, 840, 1092, 1155, 1320, 1380, 1428, 1540, 1848, 1995, 3003, 3315

Offset: 1

Vincenzo Librandi, Jun 12 2018

Intersection of A046005 and A003644. Note that A003644 = A014602 union A014603 union A192322 union {a(n)} union {5460}. - Jianing Song, Jul 12 2018

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).

Cf. A003644, A014602, A014603, A192322.

Subsequence of A046005 and A003644.

A046005 = Import["https://oeis.org/A046005/b046005.txt", "Table"][[All, 2]];
A003644 = Import["https://oeis.org/A003644/b003644.txt", "Table"][[All, 2]];
A046005 ~Intersection~ A003644 (* Jean-François Alcover, Sep 25 2019 *)

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

A316743 Discriminants of imaginary fields whose class group has exponent 2, negated.

Original entry on oeis.org

15, 20, 24, 35, 40, 51, 52, 84, 88, 91, 115, 120, 123, 132, 148, 168, 187, 195, 228, 232, 235, 267, 280, 312, 340, 372, 403, 408, 420, 427, 435, 483, 520, 532, 555, 595, 627, 660, 708, 715, 760, 795, 840, 1012, 1092, 1155, 1320, 1380, 1428, 1435, 1540, 1848, 1995, 2280, 3003, 3315, 5460

Offset: 1

Jianing Song, Jul 20 2018

This sequence lists the negated discriminants of imaginary fields whose class group is isomorphic to (C_2)^r, r > 0.
These are the negated fundamental discriminants in A133288.
Also numbers in A003644 but not in A014602. Equals A014603 union A192322 union A305416 union {5460}.

Shin-ichi Katayama, Rational Points on the Parabola and the Arithmetic of Related Algebraic Tori, J. Math. Tokushima Univ. (2024) Vol. 58, 11-31. See p. 30.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
P. J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117-124.

Cf. Negated discriminants of imaginary fields whose class group is isomorphic to (C_2)^r: A014602 (r=0), A014603 (r=1), A192322 (r=2), A305416 (r=3).

Subsequence of A003644 and A133288.

ok(n)={isfundamental(-n) && quadclassunit(-n).no > 1 && !#select(k->k<>2, quadclassunit(-n).cyc)} \\ Andrew Howroyd, Jul 20 2018

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A013658 Discriminants of imaginary quadratic fields with class number 4 (negated).

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

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A305416 Negated discriminants of imaginary quadratic number fields whose class group is isomorphic to the Klein 8-group, C_2 x C_2 x C_2.

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A316743 Discriminants of imaginary fields whose class group has exponent 2, negated.

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