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Minimal absolute value of negative discriminants of number fields of degree n.
3

%I #9 Apr 29 2021 12:06:30

%S 3,23,275,4511,9747,184607

%N Minimal absolute value of negative discriminants of number fields of degree n.

%C Conjecture: A343690(n) < a(n) for n == 0, 1 (mod 4), A343690(n) > a(n) for n == 2, 3 (mod 4).

%C 1257728 <= a(8) <= 4286875.

%H LMFDB, <a href="https://www.lmfdb.org/NumberField">Number fields</a>

%H A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/zeta.html">Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results</a>, Sem. Theorie des Nombres, Bordeaux, 2 (1990), pp. 119-141.

%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>

%F A006557(n) = Min_{A343690(n), a(n)}.

%e The number field F of degree n whose discriminant is negative and of minimal absolute value:

%e n = 2, F = Q[x]/(x^2 - x + 1), d = -3;

%e n = 3, F = Q[x]/(x^3 - x^2 + 1), d = -23;

%e n = 4, F = Q[x]/(x^4 - x^3 + 2x - 1), d = -275;

%e n = 5, F = Q[x]/(x^5 - x^3 - 2x^2 + 1), d = -4511;

%e n = 6, F = Q[x]/(x^6 - x^5 + x^4 - 2x^3 + 4x^2 - 3x + 1), d = -9747;

%e n = 7, F = Q[x]/(x^7 - x^6 - x^5 + x^4 - x^2 + x + 1), d = -184607.

%Y Cf. A343690 (the positive discriminant case), A006557 (the overall case).

%K nonn,hard,more

%O 2,1

%A _Jianing Song_, Apr 29 2021