A305474 -id:A305474 - 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, ", ")))

A305475 Constant of Hilbert class polynomial H_D(x) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Original entry on oeis.org

0, -1728, 3375, -8000, 32768, -54000, -121287375, -287496, 884736, -681472000, 12771880859375, 14670139392, 12288000, -16581375, 1566028350940383, 12167000000, -134217728000, -1790957481984, 20919104368024767633, 9103145472000, 884736000

Offset: 1

Seiichi Manyama, Jun 02 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A014601, A032354, A305474.

{a(n) = polcoeff(polclass(-2*n-n%2), 0)}

A305494 Let s(D) = Sum_{(a,b,c)} j((-b+sqrt(D))/(2a)) where (a,b,c) is taken over all the primitive reduced binary quadratic forms ax^2+bxy+cy^2 with b^2-4*ac = D. This sequence is s(D) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Original entry on oeis.org

0, 1728, -3375, 8000, -32768, 54000, -191025, 287496, -884736, 1264000, -3491750, 4834944, -12288000, 16581375, -39491307, 52250000, -117964800, 153542016, -331531596, 425692800, -884736000, 1122662608, -2257834125, 2835810000, -5541101568, 6896880000, -13136684625

Offset: 1

Seiichi Manyama, Jun 02 2018

sign

			In the case D = -15,
j((1+sqrt(-15))/2) + j((1+sqrt(-15))/4) = (-191025-85995*sqrt(5))/2 + (-191025+85995*sqrt(5))/2 = -191025.
  ----+-------------------------------------------+---------
    D | Coefficients of Hilbert class polynomial  |   a(n)
  ----+-------------------------------------------+---------
   -3 |              0,            1;             |        0
   -4 |          -1728,            1;             |     1728
   -7 |           3375,            1;             |    -3375
   -8 |          -8000,            1;             |     8000
  -11 |          32768,            1;             |   -32768
  -12 |         -54000,            1;             |    54000
  -15 |     -121287375,       191025,        1;   |  -191025
  -16 |        -287496,            1;             |   287496
  -19 |         884736,            1;             |  -884736
  -20 |     -681472000,     -1264000,        1;   |  1264000
  -23 | 12771880859375,  -5151296875,  3491750, 1;| -3491750
  -24 |    14670139392,     -4834944,        1;   |  4834944

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A014601, A032354, A305474.

cp's OEIS Frontend

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

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A305475 Constant of Hilbert class polynomial H_D(x) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

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Author

Keywords

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PARI

A305494 Let s(D) = Sum_{(a,b,c)} j((-b+sqrt(D))/(2a)) where (a,b,c) is taken over all the primitive reduced binary quadratic forms ax^2+bxy+cy^2 with b^2-4*ac = D. This sequence is s(D) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Views

Author

Keywords

Examples

Links

Crossrefs

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A305475 Constant of Hilbert class polynomial H_D(x) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A305494 Let s(D) = Sum_{(a,b,c)} j((-b+sqrt(D))/(2*a)) where (a,b,c) is taken over all the primitive reduced binary quadratic forms a*x^2+b*xy+c*y^2 with b^2-4*ac = D. This sequence is s(D) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Views

Author

Keywords

Examples

Links

Crossrefs

A305494 Let s(D) = Sum_{(a,b,c)} j((-b+sqrt(D))/(2a)) where (a,b,c) is taken over all the primitive reduced binary quadratic forms ax^2+bxy+cy^2 with b^2-4*ac = D. This sequence is s(D) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .