A391531 Discriminants of real quadratic fields with class group C_2 and form class group C_2 X C_2.

60, 105, 120, 140, 156, 165, 168, 204, 220, 264, 273, 280, 285, 312, 345, 348, 357, 364, 380, 385, 408, 429, 440, 444, 456, 460, 465, 476, 492, 552, 561, 572, 609, 616, 620, 636, 645, 665, 696, 705, 728, 732, 741, 744, 748, 760, 805, 812, 860, 861, 885, 888, 920, 952, 957, 969, 984, 988

Offset

1,1

Comments

Let Cl+(D) and Cl(D) be the narrow class group and the class group of the quadratic order of discriminant D. Then Cl+(D)/Cl(D) = 1 if D < 0 or D > 0 and the fundamental unit has norm -1, C_2 if D > 0 and the fundamental unit has norm 1.

For fundamental discriminants D, the 2-rank of Cl+(D) is omega(D) - 1, omega = A001221. Hence

Cl(D) = C_2 and Cl+(D) = C_2 X C_2 <=> Cl(D) = C_2 and omega(D) = 3.

Each term has a prime factor congruent to 3 modulo 4; see A391439 for a proof.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.

Prog

(PARI) isA391531(D) = D>1 && isfundamental(D) && qfbclassno(D)==2 && omega(D)==3

Crossrefs

Fundamental discriminants D of real quadratic fields such that

Cl(D) trivial, Cl+(D) trivial: A003656;

Cl(D) trivial, Cl+(D) = C_2: A327297;

Cl(D) = C_2, Cl+(D) = C_2: A391529;

Cl(D) = C_2, Cl+(D) = C_4: A391530;

Cl(D) = C_2, Cl+(D) = C_2 X C_2: this sequence;

Cl(D) = C_3, Cl+(D) = C_3: A391421;

Cl(D) = C_3, Cl+(D) = C_6: A391532.

Sequence in context: A267434 A048934 A097714 * A272515 A043221 A039398

Adjacent sequences: A391528 A391529 A391530 * A391532 A391533 A391534

Keyword

nonn

Author

Jianing Song, Dec 12 2025

Status

approved