Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(32133168345088\)\(\medspace = 2^{12} \cdot 1987^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.127168.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_{6}$ |
Parity: | even |
Determinant: | 1.7948.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.127168.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - x^{4} + 4x^{3} - 2x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 13 a + 28 + \left(25 a + 4\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(23 a + 19\right)\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 16 a + 6 + \left(3 a + 3\right)\cdot 29 + \left(10 a + 2\right)\cdot 29^{2} + \left(5 a + 3\right)\cdot 29^{3} + \left(28 a + 5\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 10 a + 26 + \left(4 a + 9\right)\cdot 29 + \left(8 a + 19\right)\cdot 29^{2} + \left(12 a + 19\right)\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 19 a + 18 + \left(24 a + 21\right)\cdot 29 + \left(20 a + 26\right)\cdot 29^{2} + \left(16 a + 14\right)\cdot 29^{3} + \left(28 a + 23\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 18 a + 4 + \left(16 a + 20\right)\cdot 29 + \left(21 a + 21\right)\cdot 29^{2} + \left(18 a + 7\right)\cdot 29^{3} + \left(8 a + 2\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 6 }$ | $=$ | \( 11 a + 7 + \left(12 a + 27\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(10 a + 21\right)\cdot 29^{3} + \left(20 a + 26\right)\cdot 29^{4} +O(29^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
$1$ | $1$ | $()$ | $9$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$40$ | $3$ | $(1,2,3)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)$ | $-1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.