A343690 Minimal value of positive discriminants of number fields of degree n.

1, 5, 49, 117, 1609, 28037, 612233, 1257728

Offset

1,2

Comments

Conjecture: a(n) < A343772(n) for n == 0, 1 (mod 4), a(n) > A343772(n) for n == 2, 3 (mod 4).

Links

Table of n, a(n) for n=1..8.

LMFDB, Number fields

A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results, Sem. Theorie des Nombres, Bordeaux, 2 (1990), pp. 119-141.

Index entries for sequences related to quadratic fields

Formula

A006557(n) = Min_{a(n), A343772(n)}.

Example

The number field F of degree n whose discriminant is positive and of minimal value:
n = 2, F = Q[x]/(x^2 - x - 1), d = 5;
n = 3, F = Q[x]/(x^3 - x^2 - 2x + 1), d = 49;
n = 4, F = Q[x]/(x^4 - x^3 - x^2 + x + 1), d = 117;
n = 5, F = Q[x]/(x^5 - x^3 - x^2 + x + 1), d = 1609;
n = 6, F = Q[x]/(x^6 - 2x^5 + 3x^3 - 2x - 1), d = 28037;
n = 7, F = Q[x]/(x^7 - x^6 + x^5 - x^3 + x^2 - x - 1), d = 612233;
n = 8, F = Q[x]/(x^8 - 2x^7 + 4x^5 - 4x^4 + 3x^2 - 2x + 1), d = 1257728.

Crossrefs

Cf. A343772 (the negative discriminant case), A006557 (the overall case).

Sequence in context: A298580 A299572 A266127 * A298394 A299512 A183333

Adjacent sequences: A343687 A343688 A343689 * A343691 A343692 A343693

Keyword

nonn,hard,more

Author

Jianing Song, Apr 26 2021

Status

approved