A329182 -id:A329182 - 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, ", ")))

A328825 Negative discriminants with form class group isomorphic to C_3 (negated).

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 92, 107, 108, 124, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset: 1

Jianing Song, Dec 05 2019

Also negative discriminants with form class number 3.
Conjecture: this sequence is finite and this is the full list.
The fundamental terms are listed in A006203, and that is a full sequence.
From Jianing Song, May 17 2021: (Start)
Equivalently, negative discriminants of orders whose class group is isomorphic to C_3 (negated).
The known even terms are all congruent to 12 modulo 16. Among the known even terms, k/4 is either here or in A133675. What's the reason for that?
Among the known terms, k is in A023679 if and only if k is in this sequence and k/4 is not. Is there a connection between these two sequences? (End)

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), this sequence (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), A330219 (isomorphic to C_4).

Cf. A006203, A023679.

isA328825(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[3] \\ Corrected by 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

cp's OEIS Frontend

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

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A328825 Negative discriminants with form class group isomorphic to C_3 (negated).

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

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