Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(423417472000\)\(\medspace = 2^{10} \cdot 5^{3} \cdot 149^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.39430752080.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 16T1294 |
Parity: | odd |
Determinant: | 1.2980.2t1.a.a |
Projective image: | $S_4\wr C_2$ |
Projective stem field: | Galois closure of 8.2.39430752080.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 3x^{6} + 9x^{5} - 34x^{4} + 35x^{3} - 31x^{2} + 17x - 5 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( a + \left(14 a + 36\right)\cdot 53 + 4\cdot 53^{2} + \left(18 a + 9\right)\cdot 53^{3} + \left(28 a + 31\right)\cdot 53^{4} + \left(45 a + 9\right)\cdot 53^{5} + \left(7 a + 45\right)\cdot 53^{6} + \left(30 a + 8\right)\cdot 53^{7} + \left(24 a + 11\right)\cdot 53^{8} + \left(30 a + 52\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 2 }$ | $=$ | \( 52 a + 4 + \left(38 a + 38\right)\cdot 53 + \left(52 a + 44\right)\cdot 53^{2} + \left(34 a + 27\right)\cdot 53^{3} + \left(24 a + 20\right)\cdot 53^{4} + \left(7 a + 4\right)\cdot 53^{5} + \left(45 a + 31\right)\cdot 53^{6} + \left(22 a + 15\right)\cdot 53^{7} + \left(28 a + 26\right)\cdot 53^{8} + \left(22 a + 43\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 3 }$ | $=$ | \( 16 + 8\cdot 53 + 44\cdot 53^{2} + 36\cdot 53^{3} + 7\cdot 53^{4} + 44\cdot 53^{5} + 3\cdot 53^{6} + 49\cdot 53^{7} + 47\cdot 53^{8} + 52\cdot 53^{9} +O(53^{10})\) |
$r_{ 4 }$ | $=$ | \( 25 a + 28 + \left(9 a + 9\right)\cdot 53 + \left(26 a + 7\right)\cdot 53^{2} + \left(29 a + 15\right)\cdot 53^{3} + \left(5 a + 4\right)\cdot 53^{4} + \left(40 a + 48\right)\cdot 53^{5} + \left(42 a + 28\right)\cdot 53^{6} + \left(36 a + 41\right)\cdot 53^{7} + \left(43 a + 44\right)\cdot 53^{8} + \left(9 a + 33\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 5 }$ | $=$ | \( 33 + 16\cdot 53 + 48\cdot 53^{2} + 21\cdot 53^{3} + 30\cdot 53^{4} + 24\cdot 53^{5} + 32\cdot 53^{6} + 44\cdot 53^{7} + 8\cdot 53^{8} + 3\cdot 53^{9} +O(53^{10})\) |
$r_{ 6 }$ | $=$ | \( 28 a + 22 + \left(43 a + 22\right)\cdot 53 + \left(26 a + 49\right)\cdot 53^{2} + 23 a\cdot 53^{3} + \left(47 a + 50\right)\cdot 53^{4} + \left(12 a + 43\right)\cdot 53^{5} + 10 a\cdot 53^{6} + \left(16 a + 40\right)\cdot 53^{7} + \left(9 a + 23\right)\cdot 53^{8} + \left(43 a + 29\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 7 }$ | $=$ | \( 41 a + \left(27 a + 32\right)\cdot 53 + \left(10 a + 52\right)\cdot 53^{2} + \left(24 a + 6\right)\cdot 53^{3} + \left(41 a + 16\right)\cdot 53^{4} + \left(18 a + 28\right)\cdot 53^{5} + \left(30 a + 36\right)\cdot 53^{6} + \left(17 a + 12\right)\cdot 53^{7} + \left(51 a + 10\right)\cdot 53^{8} + \left(39 a + 50\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 8 }$ | $=$ | \( 12 a + 5 + \left(25 a + 49\right)\cdot 53 + \left(42 a + 13\right)\cdot 53^{2} + \left(28 a + 40\right)\cdot 53^{3} + \left(11 a + 51\right)\cdot 53^{4} + \left(34 a + 8\right)\cdot 53^{5} + \left(22 a + 33\right)\cdot 53^{6} + \left(35 a + 52\right)\cdot 53^{7} + \left(a + 38\right)\cdot 53^{8} + \left(13 a + 52\right)\cdot 53^{9} +O(53^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(3,7)(5,8)$ | $-3$ |
$9$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $1$ |
$12$ | $2$ | $(1,2)$ | $3$ |
$24$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $3$ |
$36$ | $2$ | $(1,2)(3,8)$ | $1$ |
$36$ | $2$ | $(1,2)(3,7)(5,8)$ | $-1$ |
$16$ | $3$ | $(2,6,4)$ | $0$ |
$64$ | $3$ | $(2,6,4)(5,8,7)$ | $0$ |
$12$ | $4$ | $(3,5,7,8)$ | $-3$ |
$36$ | $4$ | $(1,2,4,6)(3,5,7,8)$ | $1$ |
$36$ | $4$ | $(1,2,4,6)(3,7)(5,8)$ | $1$ |
$72$ | $4$ | $(1,3,4,7)(2,5,6,8)$ | $-1$ |
$72$ | $4$ | $(1,2)(3,5,7,8)$ | $-1$ |
$144$ | $4$ | $(1,3,2,8)(4,5)(6,7)$ | $1$ |
$48$ | $6$ | $(2,4,6)(3,7)(5,8)$ | $0$ |
$96$ | $6$ | $(2,6,4)(3,5)$ | $0$ |
$192$ | $6$ | $(1,3)(2,8,6,7,4,5)$ | $0$ |
$144$ | $8$ | $(1,3,2,5,4,7,6,8)$ | $-1$ |
$96$ | $12$ | $(2,6,4)(3,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.