A391529 Discriminants of real quadratic fields with class group C_2 and form class group C_2.

40, 65, 85, 104, 185, 232, 265, 296, 365, 424, 481, 485, 488, 493, 533, 565, 629, 685, 808, 865, 872, 949, 965, 1037, 1073, 1157, 1165, 1189, 1192, 1241, 1256, 1261, 1285, 1385, 1417, 1448, 1465, 1565, 1576, 1585, 1649, 1685, 1769, 1781, 1832, 1853, 1865, 1921, 1985, 2117, 2152, 2165, 2173

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. So each term has exactly 2 distinct prime factors. Actually, terms are of the form 8*p for primes p == 5 (mod 8) or p*q for primes p,q == 1 (mod 4) such that (p/q) = (q/p) = -1, where (p/q) is the Legendre symbol. 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) isA391529(D) = D>1 && isfundamental(D) && qfbclassno(D)==2 && quadunitnorm(D)==-1 \\ quadunitnorm() requires PARI-GP of version 2.15 or higher

Crossrefs

Cf. A014077 (norms of fundamental units of real quadratic fields).

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: this sequence;

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: A391532.

Sequence in context: A204747 A205066 A232870 * A338373 A355778 A235283

Adjacent sequences: A391526 A391527 A391528 * A391530 A391531 A391532

Keyword

nonn

Author

Jianing Song, Dec 12 2025

Status

approved