

A343290


Irregular triangle read by rows: For n >= 1, 0 <= k <= floor(n/2), T(n,k) is the minimal discriminant (in absolute value) of the number fields with signature r_1 = n  2*k, r_2 = k.


1



1, 5, 3, 49, 23, 725, 275, 117, 14641, 4511, 1609, 300125, 92779, 28037, 9747, 20134393, 2306599, 612233, 184607, 282300416
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OFFSET

1,2


COMMENTS

For a number field F with degree n, the signature of F is a pair of numbers (r_1, r_2), where r_1 is the number of real embeddings of F, r_2 is half the number of complex embeddings of F. Obviously, we have r_1 + 2*r_2 = n. For example, a real quadratic field has r_1 = 2, r_2 = 0, and an imaginary quadratic field has r_1 = 0, r_2 = 1.
T(0,4) = 1257728, T(9,0) = 9685993193.
The sign of T(n,k) is (1)^k.
It seems that the terms of each row are strictly decreasing in absolute value.


LINKS

Table of n, a(n) for n=1..20.
LMFDB, Number fields


FORMULA

A006557(n) = Min_{k=1..floor(n/2)} T(n,k).
A343690(n) = Min_{k=1..floor(n/2), k even} T(n,k).
A343772(n) = Min_{k=1..floor(n/2), k odd} T(n,k), n >= 2.


EXAMPLE

Let F be a field with signature r_1 = 5, r_2 = 0, then disc(F) >= 14641. The equality holds when F = Q[x]/(x^5  x^4  4x^3 + 3x^2 + 3^x  1), so T(5,0) = 14641.
Let F be a field with signature r_1 = 3, r_2 = 1, then disc(F) <= 4511. The equality holds when F = Q[x]/(x^5  x^3  2x^2 + 1), so T(5,1) = 4511.
Let F be a field with signature r_1 = 7, r_2 = 0, then disc(F) >= 20134393. The equality holds when F = Q[x]/(x7  x^6  6x^5 + 4x^4 + 10x^3  4x^2  4x + 1), so T(7,0) = 20134393.


CROSSREFS

Cf. A006557, A343690, A343772.
First column is A006554. Second column is A006555 (negated).
Sequence in context: A187278 A288184 A323779 * A027858 A181755 A007299
Adjacent sequences: A343287 A343288 A343289 * A343291 A343292 A343293


KEYWORD

sign,hard,more


AUTHOR

Jianing Song, Apr 28 2021


STATUS

approved



