Basic invariants
Dimension: | $10$ |
Group: | $S_6$ |
Conductor: | \(2888816545234944\)\(\medspace = 2^{26} \cdot 3^{16} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.13436928.5 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T164 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.13436928.5 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 6x^{4} - 4x^{3} + 6x^{2} - 6 \) . |
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:
$r_{ 1 }$ | $=$ | \( 125 + 13\cdot 167 + 71\cdot 167^{2} + 8\cdot 167^{3} + 52\cdot 167^{4} +O(167^{5})\) |
$r_{ 2 }$ | $=$ | \( 141 + 151\cdot 167 + 21\cdot 167^{2} + 13\cdot 167^{3} + 133\cdot 167^{4} +O(167^{5})\) |
$r_{ 3 }$ | $=$ | \( 67 a + 128 + \left(164 a + 103\right)\cdot 167 + \left(112 a + 44\right)\cdot 167^{2} + \left(59 a + 111\right)\cdot 167^{3} + \left(a + 25\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 4 }$ | $=$ | \( 100 a + 28 + \left(2 a + 34\right)\cdot 167 + \left(54 a + 160\right)\cdot 167^{2} + \left(107 a + 57\right)\cdot 167^{3} + \left(165 a + 134\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 5 }$ | $=$ | \( 131 a + 141 + \left(44 a + 141\right)\cdot 167 + \left(132 a + 57\right)\cdot 167^{2} + \left(58 a + 108\right)\cdot 167^{3} + \left(17 a + 98\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 6 }$ | $=$ | \( 36 a + 105 + \left(122 a + 55\right)\cdot 167 + \left(34 a + 145\right)\cdot 167^{2} + \left(108 a + 34\right)\cdot 167^{3} + \left(149 a + 57\right)\cdot 167^{4} +O(167^{5})\) |
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 |
$1$ | $1$ | $()$ | $10$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
$15$ | $2$ | $(1,2)$ | $2$ |
$45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$40$ | $3$ | $(1,2,3)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.