A379019 Positive integers k such that the simplest cubic field defined by x^3 - k*x^2 - (k+3)*x - 1 is not monogenic.

21, 30, 41, 48, 57, 75, 84, 90, 100, 102, 103, 111, 129, 138, 139, 152, 154, 156, 165, 183, 188, 192, 201, 204, 210, 219, 235, 237, 246, 250, 264, 269, 271, 273, 291, 299, 300, 318, 327, 335, 345, 348, 354, 356, 372, 374, 381, 384, 398, 399, 404, 408, 426, 433, 435, 438, 446, 453, 462, 480

Offset: 1

Author

Robin Visser, Dec 13 2024

Keywords

• nonn

Comments

These are the positive integers k such that the ring of integers O_K of the simplest cubic field K = Q[x]/(x^3 - k*x^2 - (k+3)*x - 1) does not have a power integral basis of the form {1, a, a^2} for any element a in O_K.

Links

• D. Gil-Muñoz and M. Tinková, Additive structure of non-monogenic simplest cubic fields, arXiv:2212.00364 [math.NT], 2022.
• T. Kashio and R. Sekigawa, The characterization of cyclic cubic fields with power integral bases, Kodai Math. J. 44 (2021), no. 2, 290-306.
• D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.

Crossrefs

Cf. A005472.

Programs

Magma

is_A379019 := function(k)
    R := PolynomialRing(Integers());
    K := NumberField(x^3 - k*x^2 - (k+3)*x - 1);
    return #IndexFormEquation(MaximalOrder(K), 1) eq 0;
end function;
[k : k in [1..1000] | is_A379019(k)];