A278791 -id:A278791 - cp's OEIS frontend

A023679 Discriminants of complex cubic fields (negated).

23, 31, 44, 59, 76, 83, 87, 104, 107, 108, 116, 135, 139, 140, 152, 172, 175, 199, 200, 204, 211, 212, 216, 231, 239, 243, 244, 247, 255, 268, 283, 300, 307, 324, 327, 331, 335, 339, 351, 356, 364, 367, 379, 411, 419, 424, 431, 436, 439, 440, 451, 459, 460, 472, 484, 491, 492, 499, 503

Offset: 1

N. J. A. Sloane

nonn

			The field Q[x]/(x^3 - x^2 + 1) is the complex cubic field with the smallest absolute discriminant of 23. - _Robin Visser_, Mar 27 2025

M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, p. 437.

Robin Visser, Table of n, a(n) for n = 1..10000 (taken from the Jones-Roberts database)
J. W. Jones and D. P. Roberts, A database of number fields, LMS J. Comput. Math. 17 (2014), no. 1, 595-618.

Cf. A006832, A187773, A278791.

More terms added by Robin Visser, Mar 27 2025, taken from the database of John Jones and David Roberts.

A278790 Number of real cubic fields with discriminant <= 10^n.

Original entry on oeis.org

0, 2, 27, 382, 4804, 54600, 592922, 6248290, 64659361, 661448081, 6715824025

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 real cubic fields with discriminant <= X is asymptotic to X/(12*zeta(3)) = (0.069325...)*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. A006832, A278791.

cp's OEIS Frontend

A023679 Discriminants of complex cubic fields (negated).

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