Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(192\!\cdots\!649\)\(\medspace = 73^{6} \cdot 709^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.51757.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T145 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.51757.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{3} - 2x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: \( x^{2} + 101x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 23 a + 80 + \left(43 a + 75\right)\cdot 113 + \left(104 a + 69\right)\cdot 113^{2} + \left(a + 98\right)\cdot 113^{3} + \left(29 a + 104\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 2 }$ | $=$ | \( 74 a + 93 + \left(81 a + 50\right)\cdot 113 + \left(61 a + 28\right)\cdot 113^{2} + \left(101 a + 41\right)\cdot 113^{3} + \left(7 a + 22\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 3 }$ | $=$ | \( 13 a + 71 + \left(5 a + 52\right)\cdot 113 + \left(54 a + 58\right)\cdot 113^{2} + \left(96 a + 13\right)\cdot 113^{3} + \left(96 a + 74\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 4 }$ | $=$ | \( 39 a + 77 + \left(31 a + 52\right)\cdot 113 + \left(51 a + 9\right)\cdot 113^{2} + \left(11 a + 68\right)\cdot 113^{3} + \left(105 a + 15\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 5 }$ | $=$ | \( 90 a + 17 + \left(69 a + 6\right)\cdot 113 + \left(8 a + 36\right)\cdot 113^{2} + \left(111 a + 17\right)\cdot 113^{3} + \left(83 a + 112\right)\cdot 113^{4} +O(113^{5})\) |
$r_{ 6 }$ | $=$ | \( 100 a + 1 + \left(107 a + 101\right)\cdot 113 + \left(58 a + 23\right)\cdot 113^{2} + \left(16 a + 100\right)\cdot 113^{3} + \left(16 a + 9\right)\cdot 113^{4} +O(113^{5})\) |
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.