Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(828465164288\)\(\medspace = 2^{12} \cdot 587^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.37568.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_{6}$ |
Parity: | even |
Determinant: | 1.2348.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.37568.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 2x^{4} - x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: \( x^{2} + 166x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 21 a + 3 + \left(150 a + 82\right)\cdot 167 + \left(89 a + 154\right)\cdot 167^{2} + \left(128 a + 73\right)\cdot 167^{3} + \left(142 a + 128\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 2 }$ | $=$ | \( 85 a + 8 + \left(67 a + 69\right)\cdot 167 + \left(33 a + 122\right)\cdot 167^{2} + \left(97 a + 61\right)\cdot 167^{3} + \left(128 a + 31\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 3 }$ | $=$ | \( 146 a + 24 + \left(16 a + 44\right)\cdot 167 + \left(77 a + 94\right)\cdot 167^{2} + \left(38 a + 112\right)\cdot 167^{3} + \left(24 a + 142\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 4 }$ | $=$ | \( 31 a + 5 + \left(60 a + 29\right)\cdot 167 + \left(104 a + 82\right)\cdot 167^{2} + \left(106 a + 62\right)\cdot 167^{3} + \left(58 a + 8\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 5 }$ | $=$ | \( 82 a + 93 + \left(99 a + 51\right)\cdot 167 + \left(133 a + 88\right)\cdot 167^{2} + \left(69 a + 125\right)\cdot 167^{3} + \left(38 a + 62\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 6 }$ | $=$ | \( 136 a + 36 + \left(106 a + 58\right)\cdot 167 + \left(62 a + 126\right)\cdot 167^{2} + \left(60 a + 64\right)\cdot 167^{3} + \left(108 a + 127\right)\cdot 167^{4} +O(167^{5})\) |
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.