cp's OEIS Frontend

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

%I A379019 #7 Dec 16 2024 08:53:50
%S A379019 21,30,41,48,57,75,84,90,100,102,103,111,129,138,139,152,154,156,165,
%T A379019 183,188,192,201,204,210,219,235,237,246,250,264,269,271,273,291,299,
%U A379019 300,318,327,335,345,348,354,356,372,374,381,384,398,399,404,408,426,433,435,438,446,453,462,480
%N A379019 Positive integers k such that the simplest cubic field defined by x^3 - k*x^2 - (k+3)*x - 1 is not monogenic.
%C A379019 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.
%H A379019 D. Gil-Muñoz and M. Tinková, <a href="https://arxiv.org/abs/2212.00364">Additive structure of non-monogenic simplest cubic fields</a>, arXiv:2212.00364 [math.NT], 2022.
%H A379019 T. Kashio and R. Sekigawa, <a href="https://doi.org/10.2996/kmj44204">The characterization of cyclic cubic fields with power integral bases</a>, Kodai Math. J. 44 (2021), no. 2, 290-306.
%H A379019 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8">The simplest cubic fields</a>, Math. Comp., 28 (1974), 1137-1152.
%o A379019 (Magma)
%o A379019 is_A379019 := function(k)
%o A379019     R<x> := PolynomialRing(Integers());
%o A379019     K<a> := NumberField(x^3 - k*x^2 - (k+3)*x - 1);
%o A379019     return #IndexFormEquation(MaximalOrder(K), 1) eq 0;
%o A379019 end function;
%o A379019 [k : k in [1..1000] | is_A379019(k)];
%Y A379019 Cf. A005472.
%K A379019 nonn
%O A379019 1,1
%A A379019 _Robin Visser_, Dec 13 2024