Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(116095057920000\)\(\medspace = 2^{20} \cdot 3^{11} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.3583180800.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T274 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.3583180800.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - 8x^{5} + 6x^{4} - 2x^{2} - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$:
\( x^{2} + 103x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 82 a + 20 + \left(61 a + 10\right)\cdot 107 + \left(10 a + 30\right)\cdot 107^{2} + \left(7 a + 92\right)\cdot 107^{3} + \left(66 a + 96\right)\cdot 107^{4} + \left(72 a + 49\right)\cdot 107^{5} + \left(65 a + 22\right)\cdot 107^{6} + \left(14 a + 83\right)\cdot 107^{7} + \left(53 a + 93\right)\cdot 107^{8} + \left(35 a + 10\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 101 + \left(100 a + 64\right)\cdot 107 + \left(27 a + 30\right)\cdot 107^{2} + \left(21 a + 21\right)\cdot 107^{3} + \left(45 a + 36\right)\cdot 107^{4} + \left(106 a + 101\right)\cdot 107^{5} + \left(105 a + 71\right)\cdot 107^{6} + \left(49 a + 55\right)\cdot 107^{7} + \left(70 a + 12\right)\cdot 107^{8} + \left(44 a + 84\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 + 81\cdot 107 + 94\cdot 107^{2} + 38\cdot 107^{3} + 76\cdot 107^{4} + 30\cdot 107^{5} + 64\cdot 107^{6} + 85\cdot 107^{7} + 96\cdot 107^{8} + 101\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 a + 37 + \left(65 a + 73\right)\cdot 107 + \left(68 a + 55\right)\cdot 107^{2} + \left(55 a + 81\right)\cdot 107^{3} + \left(74 a + 64\right)\cdot 107^{4} + \left(84 a + 18\right)\cdot 107^{5} + \left(94 a + 74\right)\cdot 107^{6} + \left(103 a + 43\right)\cdot 107^{7} + \left(46 a + 102\right)\cdot 107^{8} + \left(103 a + 8\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 94 + 18\cdot 107 + 6\cdot 107^{2} + 91\cdot 107^{3} + 45\cdot 107^{4} + 44\cdot 107^{5} + 80\cdot 107^{6} + 104\cdot 107^{7} + 41\cdot 107^{8} + 89\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 101 a + 18 + \left(6 a + 31\right)\cdot 107 + \left(79 a + 42\right)\cdot 107^{2} + \left(85 a + 78\right)\cdot 107^{3} + \left(61 a + 88\right)\cdot 107^{4} + 53\cdot 107^{5} + \left(a + 68\right)\cdot 107^{6} + \left(57 a + 42\right)\cdot 107^{7} + \left(36 a + 30\right)\cdot 107^{8} + \left(62 a + 85\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 a + 27 + \left(45 a + 68\right)\cdot 107 + \left(96 a + 10\right)\cdot 107^{2} + \left(99 a + 3\right)\cdot 107^{3} + \left(40 a + 33\right)\cdot 107^{4} + \left(34 a + 60\right)\cdot 107^{5} + \left(41 a + 105\right)\cdot 107^{6} + \left(92 a + 75\right)\cdot 107^{7} + \left(53 a + 77\right)\cdot 107^{8} + \left(71 a + 99\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 66 a + 94 + \left(41 a + 79\right)\cdot 107 + \left(38 a + 50\right)\cdot 107^{2} + \left(51 a + 21\right)\cdot 107^{3} + \left(32 a + 93\right)\cdot 107^{4} + \left(22 a + 68\right)\cdot 107^{5} + \left(12 a + 47\right)\cdot 107^{6} + \left(3 a + 43\right)\cdot 107^{7} + \left(60 a + 79\right)\cdot 107^{8} + \left(3 a + 54\right)\cdot 107^{9} +O(107^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $-3$ |
| $9$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-3$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,7)(2,3)(4,8)$ | $-1$ |
| $16$ | $3$ | $(3,5,6)$ | $0$ |
| $64$ | $3$ | $(2,5,6)(4,7,8)$ | $0$ |
| $12$ | $4$ | $(2,3,5,6)$ | $-3$ |
| $36$ | $4$ | $(1,4,7,8)(2,3,5,6)$ | $1$ |
| $36$ | $4$ | $(1,4,7,8)(2,5)(3,6)$ | $1$ |
| $72$ | $4$ | $(1,2,7,5)(3,8,6,4)$ | $1$ |
| $72$ | $4$ | $(1,4,7,8)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,4,3)(5,7)(6,8)$ | $-1$ |
| $48$ | $6$ | $(1,7)(3,6,5)(4,8)$ | $0$ |
| $96$ | $6$ | $(2,3)(4,8,7)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,4,5,7,6,8)$ | $0$ |
| $144$ | $8$ | $(1,2,4,3,7,5,8,6)$ | $1$ |
| $96$ | $12$ | $(1,4,7,8)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.