A391532 Discriminants of real quadratic fields with class group C_3 and form class group C_6.

316, 321, 469, 473, 568, 892, 993, 1016, 1101, 1257, 1304, 1436, 1509, 1772, 1929, 1957, 2021, 2101, 2177, 2429, 2589, 2636, 2981, 3173, 3261, 3356, 3569, 3736, 3873, 3941, 3957, 3981, 4193, 4281, 4364, 4684, 4749, 4841, 4853, 4857, 4892, 5089, 5468, 5497, 5613, 5637, 5853, 5912, 6092, 6209, 6268, 6289

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_3 and Cl+(D) = C_6 <=> Cl(D) = C_3 and omega(D) = 2.

Terms are of the form 4*p, 8*p for primes p == 3 (mod 4) or p*q for primes p,q == 3 (mod 4). See Theorem 1 and Theorem 2 of Ezra Brown's link.

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) isA391532(D) = D>1 && isfundamental(D) && qfbclassno(D)==3 && !isprime(D)

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: A391531;

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

Cl(D) = C_3, Cl+(D) = C_6: this sequence.

Sequence in context: A048905 A200314 A349419 * A115559 A033861 A179155

Adjacent sequences: A391529 A391530 A391531 * A391533 A391534 A391535

Keyword

nonn

Author

Jianing Song, Dec 12 2025

Status

approved