A390079 Discriminants of orders of real quadratic fields with 1 class per genus.

5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 32, 33, 37, 40, 41, 44, 45, 48, 52, 53, 56, 57, 60, 61, 65, 68, 69, 72, 73, 76, 77, 80, 84, 85, 88, 89, 92, 93, 96, 97, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 124, 125, 128, 129, 132, 133, 137, 140, 141, 149, 152, 153, 156, 157, 160, 161, 164, 165, 168, 172, 173, 176, 177, 180, 181, 184, 185, 188, 189, 192, 193, 197, 200

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. Sequence gives D such that Cl+(D) = (C_2)^r for some r >= 0.

A391419 gives D such that Cl(D) = (C_2)^r. For D in A391419:

- If the fundamental unit has norm -1, then Cl+(D) = Cl(D), so D is in this sequence;

- If 16|D or D has a prime factor congruent to 3 modulo 4, then Cl+(D) = Cl(D) X C_2, so D is in this sequence;

- In the remaining cases, Cl+(D) has a cyclic subgroup of order 4, so D is not in this sequence.

See A391439 for more details.

Links

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

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Prog

(PARI) isA390079(n) = if(n%4<=1 && !issquare(n) && !#select(k->k<>2, quadclassunit(n).cyc), if(n%16==0 || quadunitnorm(n)==-1, return(1)); my(f = factor(n)[, 1]~); for(i=1, #f, if(f[i]%4==3, return(1)))); return(0) \\ quadunitnorm() requires PARI-GP of version 2.15 or higher

Crossrefs

Cf. A003171 (the sequence for imaginary quadratic fields).

Cf. A306638 (norms of the fundamental unit).

Sequences related to the class groups of real quadratic fields:

For class groups related to fundamental discriminants (A003658): A391436, A391437 (2-rank), A391426, A391435 (number of genera), A391417 (exponent <= 2);

For form class groups related to fundamental discriminants: A317991, A317992 (2-rank), A317989, A317990 (number of genera), A391422 (exponent <= 2);

For class groups related to all discriminants (A079896): A391439 (2-rank), A391438 (number of genera), A391419 (exponent <= 2);

For form class groups related to all discriminants: A391441 (2-rank), A391440 (number of genera), this sequence (exponent <= 2).

Sequence in context: A116602 A079896 A391419 * A133315 A003658 A391417

Adjacent sequences: A390076 A390077 A390078 * A390080 A390081 A390082

Keyword

nonn,changed

Author

Jianing Song, Nov 24 2025

Status

approved