A328825 -id:A328825 - cp's OEIS frontend

A322710 Negative discriminants with form class number 2 (negated).

15, 20, 24, 32, 35, 36, 40, 48, 51, 52, 60, 64, 72, 75, 88, 91, 99, 100, 112, 115, 123, 147, 148, 187, 232, 235, 267, 403, 427

Offset: 1

Jianing Song, Dec 24 2018

This is the full sequence.

The j-invariants for these discriminants are quadratic integers. See the links below for a full list.

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013
Jianing Song, j-invariants for the discriminants with form class number 2 and their Hilbert class polynomials
Eric Weisstein's World of Mathematics, Class Number

Cf. A014603, A305474.

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), this sequence (isomorphic to C_2), A328825 (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), A330219 (isomorphic to C_4).

for(n=1, 500, if((-n)%4<=1&&quadclassunit(-n)[1]==2, print1(n, ", ")))

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

A344409 Positive discriminants of orders with class number 3.

Original entry on oeis.org

148, 229, 257, 316, 321, 404, 469, 473, 564, 568, 592, 621, 733, 756, 761, 788, 837, 892, 916, 993, 1016, 1028, 1076, 1101, 1229, 1257, 1264, 1284, 1304, 1332, 1373, 1396, 1436, 1489, 1492, 1509, 1524, 1556, 1573, 1593, 1616, 1620, 1772, 1876, 1892, 1901, 1929, 1944

Offset: 1

Jianing Song, May 17 2021

nonn

Also positive discriminants of orders with class group isomorphic to C_3.
The fundamental terms are listed in A094612.
It seems that for most k in this sequence, 4*k is also in this sequence. The smallest k such that this is not true is k = 564.
Conjecture: if a term k is congruent to 4 modulo 16, then k/4 is either here or in A133315; if a term k is congruent to 0 modulo 16, then k/4 is in this sequence.
Conjecture: a term k is in A006832 if and only if k/4 is not in this sequence.

Jianing Song, Table of n, a(n) for n = 1..10001

Cf. A133315 (positive discriminants of orders with class number 1), A344408 (class number 2), this sequence (class number 3).

Cf. A328825 (the negative discriminant case), A094612, A006832.

isA344409(d) = (d>0) && !issquare(d) && ((d%4==0)||(d%4==1)) && quadclassunit(d)[2]==[3]

cp's OEIS Frontend

A322710 Negative discriminants with form class number 2 (negated).

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

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

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A344409 Positive discriminants of orders with class number 3.

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