A052475 - OEIS

A052475

Discriminants of real quadratic number fields K with class number 2 such that the Hilbert class field of K is K(sqrt(5)).

40, 60, 65, 85, 105, 120, 140, 165, 185, 205, 220, 265, 280, 285, 305, 345, 365, 380, 385, 440, 460, 465, 485, 545, 565, 620, 645, 665, 685, 705, 745, 760, 805, 860, 865, 885, 920, 965, 1005, 1065, 1085, 1165, 1180, 1185, 1205, 1240, 1245, 1265, 1285, 1340, 1385, 1405, 1420, 1465, 1505

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For a given number field K, the Hilbert class field of K is the maximal unramified abelian extension of K. If the class number of K is 2, then the Hilbert class field of K is a quadratic extension of K. This sequence therefore lists the possible discriminants D of real quadratic fields K with class number 2 such that K(sqrt(5))/K is unramified. - Robin Visser, Dec 19 2025

REFERENCES

Henri Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, pp. 534-535.

LINKS

Robin Visser, Table of n, a(n) for n = 1..10000

Henri Cohen and Xavier-François Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244. See page 1241.

EXAMPLE

From Robin Visser, Dec 19 2025: (Start)

a(1) = 40, as the real quadratic field K = Q(sqrt(10)) has discriminant 40 and class number 2, and the Hilbert class field of K is K(sqrt(5)) = Q(sqrt(10), sqrt(5)).

a(2) = 60, as the real quadratic field K = Q(sqrt(15)) has discriminant 60 and class number 2, and the Hilbert class field of K is K(sqrt(5)) = Q(sqrt(15), sqrt(5)).

a(3) = 65, as the real quadratic field K = Q(sqrt(65)) has discriminant 65 and class number 2, and the Hilbert class field of K is K(sqrt(5)) = Q(sqrt(65), sqrt(5)). (End)

PROG

(SageMath)

def is_A052475(k):

if Integer(k).is_square(): return False

K.<a> = QuadraticField(k)

if (K.disc() != k) or (K.class_number() != 2): return False

H.<b> = K.hilbert_class_field()

return H.is_isomorphic(K.extension(x^2-5, 'c'))

print([k for k in range(1, 2000) if is_A052475(k)]) # Robin Visser, Dec 19 2025

CROSSREFS

Cf. A035120, A003656, A052476, A052477, A052478, A052479.

Sequence in context: A344408 A035120 A094619 * A182205 A195571 A060672