A052477 - OEIS

A052477

Discriminants of real quadratic number fields K with class number 2 such that the Hilbert class field of K is K(sqrt(3)).

60, 156, 204, 348, 444, 492, 636, 732, 1068, 1212, 1308, 1356, 1644, 1788, 1884, 2076, 2172, 2364, 2748, 2796, 3084, 3228, 3324, 3372, 3516, 3804, 4188, 4236, 4476, 4668, 4812, 5052, 5388, 5532, 6252, 6492, 6684, 6828, 7116, 7356, 7692, 7836, 7932, 8412, 8508, 8796, 9084, 9132, 9276, 9564

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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(3))/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) = 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(3)) = Q(sqrt(15), sqrt(3)).

a(2) = 156, as the real quadratic field K = Q(sqrt(39)) has discriminant 156 and class number 2, and the Hilbert class field of K is K(sqrt(3)) = Q(sqrt(39), sqrt(3)).

a(3) = 204, as the real quadratic field K = Q(sqrt(51)) has discriminant 204 and class number 2, and the Hilbert class field of K is K(sqrt(3)) = Q(sqrt(51), sqrt(3)). (End)

PROG

(SageMath)

def is_A052477(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^2-3, 'c'))

print([k for k in range(1, 2000) if is_A052477(k)]) # Robin Visser, Dec 19 2025

CROSSREFS

Cf. A035120, A003656, A052475, A052476, A052478, A052479.

Sequence in context: A135199 A112065 A252018 * A044392 A044773 A256633