A316743 Discriminants of imaginary fields whose class group has exponent 2, negated.

15, 20, 24, 35, 40, 51, 52, 84, 88, 91, 115, 120, 123, 132, 148, 168, 187, 195, 228, 232, 235, 267, 280, 312, 340, 372, 403, 408, 420, 427, 435, 483, 520, 532, 555, 595, 627, 660, 708, 715, 760, 795, 840, 1012, 1092, 1155, 1320, 1380, 1428, 1435, 1540, 1848, 1995, 2280, 3003, 3315, 5460

Offset: 1

Author

Jianing Song, Jul 20 2018

Keywords

• nonn
• fini
• full

Comments

This sequence lists the negated discriminants of imaginary fields whose class group is isomorphic to (C_2)^r, r > 0.

These are the negated fundamental discriminants in A133288.

Also numbers in A003644 but not in A014602. Equals A014603 union A192322 union A305416 union {5460}.

Links

• Shin-ichi Katayama, Rational Points on the Parabola and the Arithmetic of Related Algebraic Tori, J. Math. Tokushima Univ. (2024) Vol. 58, 11-31. See p. 30.
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
• P. J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117-124.

Crossrefs

Cf. Negated discriminants of imaginary fields whose class group is isomorphic to (C_2)^r: A014602 (r=0), A014603 (r=1), A192322 (r=2), A305416 (r=3).

Subsequence of A003644 and A133288.

Programs

PARI

ok(n)={isfundamental(-n) && quadclassunit(-n).no > 1 && !#select(k->k<>2, quadclassunit(-n).cyc)} \\ Andrew Howroyd, Jul 20 2018