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Number of degree 3 number fields unramified outside the first n prime numbers.
2

%I #14 Dec 10 2023 08:25:37

%S 0,9,32,108,360,1168,3638,11492,35638,111059

%N Number of degree 3 number fields unramified outside the first n prime numbers.

%C B. Matschke showed that a(11) = 340618 assuming the Generalized Riemann Hypothesis.

%H K. Belabas, <a href="https://doi.org/10.1090/S0025-5718-97-00846-6">A fast algorithm to compute cubic fields</a>, Math. Comp. 66 (1997), no. 219, 1213-1237.

%H J. W. Jones and D. P. Roberts, <a href="https://doi.org/10.1112/S1461157014000424">A database of number fields</a>, LMS J. Comput. Math. 17 (2014), no. 1, 595-618.

%H B. Matschke, <a href="https://github.com/bmatschke/s-unit-equations/tree/main/elliptic-curve-tables/fields/numberfields-unramified-outside-S">Number fields unramified outside S</a>.

%e For n = 1, there are no cubic number fields unramified away from 2, so a(1) = 0.

%e For n = 2, the a(2) = 9 cubic number fields unramified away from {2,3} can be given by Q(a) where a is a root of x^3 - 3x - 1, x^3 - 2, x^3 + 3x - 2, x^3 - 3, x^3 - 3x - 4, x^3 - 3x - 10, x^3 - 12, x^3 - 6, or x^3 - 9x - 6.

%Y Cf. A126646 (degree 2), A368057 (degree 4).

%K nonn,more

%O 1,2

%A _Robin Visser_, Nov 26 2023