A023679 -id:A023679 - 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.

A328825 Negative discriminants with form class group isomorphic to C_3 (negated).

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 92, 107, 108, 124, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset: 1

Jianing Song, Dec 05 2019

Also negative discriminants with form class number 3.
Conjecture: this sequence is finite and this is the full list.
The fundamental terms are listed in A006203, and that is a full sequence.
From Jianing Song, May 17 2021: (Start)
Equivalently, negative discriminants of orders whose class group is isomorphic to C_3 (negated).
The known even terms are all congruent to 12 modulo 16. Among the known even terms, k/4 is either here or in A133675. What's the reason for that?
Among the known terms, k is in A023679 if and only if k is in this sequence and k/4 is not. Is there a connection between these two sequences? (End)

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

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), A322710 (isomorphic to C_2), this sequence (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), A330219 (isomorphic to C_4).

Cf. A006203, A023679.

isA328825(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[3] \\ Corrected by Jianing Song, May 17 2021

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.

A106312 Possible negative values of the discriminant of irreducible monic integral cubic polynomials.

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 87, 104, 107, 108, 116, 135, 139, 140, 152, 172, 175, 176, 199, 200, 204, 211, 212, 216, 231, 236, 239, 243, 244, 247, 255, 268, 279, 283, 300, 304, 307, 324, 327, 331, 332, 335, 339, 351, 356, 364, 367, 379, 411, 416, 419, 424, 428

Offset: 1

T. D. Noe, May 17 2005

nonn

These discriminants were found by examining all irreducible integral cubic polynomials x^3+ax^2+bx+c for a,b,c in [ -30,30]. Closely related to A023679, discriminants of complex cubic fields.

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
Eric Weisstein's World of Mathematics, Polynomial Discriminant

A187773 Discriminants of norm-Euclidean complex cubic fields (negated).

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 87, 104, 107, 108, 116, 135, 139, 140, 152, 172, 175, 200, 204, 211, 212, 216, 231, 239, 243, 244, 247, 255, 268, 300, 324, 356, 379, 411, 419, 424, 431, 440, 451, 460, 472, 484, 492, 499, 503, 515, 516, 519, 543, 628, 652, 687, 696, 728, 744, 771, 815, 876

Offset: 1

Arkadiusz Wesolowski, Jan 05 2013

D. A. Clark, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta math. 83 (1994), pp. 327-330.

Cf. A023679.

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A006832 Discriminants of totally real cubic fields.

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A328825 Negative discriminants with form class group isomorphic to C_3 (negated).

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

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A106312 Possible negative values of the discriminant of irreducible monic integral cubic polynomials.

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A187773 Discriminants of norm-Euclidean complex cubic fields (negated).

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