A278790 -id:A278790 - cp's OEIS frontend

A006832 Discriminants of totally real cubic fields.

49, 81, 148, 169, 229, 257, 316, 321, 361, 404, 469, 473, 564, 568, 621, 697, 733, 756, 761, 785, 788, 837, 892, 940, 961, 985, 993, 1016, 1076, 1101, 1129, 1229, 1257, 1300, 1304, 1345, 1369, 1373, 1384, 1396, 1425, 1436, 1489, 1492, 1509, 1524

Offset: 1

N. J. A. Sloane

nonn

			The field Q[x]/(x^3 - x^2 - 2*x + 1) is the totally real cubic field with the smallest discriminant of 49. - _Robin Visser_, Apr 17 2025

Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 436.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..130 from R. J. Mathar).
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213-1237.
T. W. Cusick and L. Schoenfeld, A table of fundamental pairs of units in totally real cubic fields, Math. Comp. 48 (1987), 147-158.
V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495-518.
P. Llorente and J. Quer, On totally real cubic fields with discriminant D < 10^7, Math. Comp. 50 (1988), no. 182, 581-594.

Cf. A023679, A278790.

A278791 Number of complex cubic fields with discriminant >= -10^n.

Original entry on oeis.org

0, 7, 127, 1520, 17041, 182417, 1905514, 19609185, 199884780, 2024660098, 20422230540

Offset: 1

Christopher E. Thompson, Nov 28 2016

Belabas invented an algorithm to identify all cubic fields with a discriminant bounded by X in essentially linear time, and computed the above values up to a(11).

The number of complex cubic fields with discriminant >= -X is asymptotic to X/(4*zeta(3)) = (0.207976...)*X. The second order term was conjectured by Roberts to be a known constant times X^{5/6}, and this was subsequently proved by Bhargava et al.

Henri Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 426 (and Chapter 8 more generally)

Karim Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237.
Manjul Bhargava, Arul Shankar, Jacob Tsimerman, On the Davenport-Heilbronn theorems and second order terms, Invent. math. 193:2 (2013) 439-499.
David P. Roberts, Density of cubic field discriminants, Math. Comp. 70 (2001), 1699-1705.

Cf. A023679, A278790.

cp's OEIS Frontend

A006832 Discriminants of totally real cubic fields.

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