Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(6553672664427\)\(\medspace = 3^{4} \cdot 7^{4} \cdot 17^{3} \cdot 19^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.699822459.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.323.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.699822459.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 3x^{6} - 10x^{3} + 17x^{2} - 11x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{3} + 3x + 81 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 81 a^{2} + 53 a + 65 + \left(48 a^{2} + 25 a + 41\right)\cdot 83 + \left(81 a^{2} + 12 a + 23\right)\cdot 83^{2} + \left(25 a^{2} + 71\right)\cdot 83^{3} + \left(69 a^{2} + 16 a + 9\right)\cdot 83^{4} + \left(32 a^{2} + 59 a + 1\right)\cdot 83^{5} + \left(71 a^{2} + 15 a + 58\right)\cdot 83^{6} + \left(34 a^{2} + 53 a + 3\right)\cdot 83^{7} + \left(41 a^{2} + 70 a + 24\right)\cdot 83^{8} + \left(17 a^{2} + 35 a + 70\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 17\cdot 83 + 16\cdot 83^{2} + 23\cdot 83^{3} + 45\cdot 83^{4} + 33\cdot 83^{5} + 34\cdot 83^{6} + 79\cdot 83^{7} + 29\cdot 83^{8} + 3\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 54\cdot 83 + 6\cdot 83^{2} + 65\cdot 83^{3} + 41\cdot 83^{4} + 66\cdot 83^{5} + 53\cdot 83^{6} + 32\cdot 83^{7} + 26\cdot 83^{8} + 35\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 a^{2} + 23 a + 15 + \left(13 a^{2} + 54 a + 4\right)\cdot 83 + \left(3 a^{2} + 67 a\right)\cdot 83^{2} + \left(67 a^{2} + 47 a + 30\right)\cdot 83^{3} + \left(67 a^{2} + 14 a + 69\right)\cdot 83^{4} + \left(49 a^{2} + 46 a + 47\right)\cdot 83^{5} + \left(46 a^{2} + 40 a + 65\right)\cdot 83^{6} + \left(53 a^{2} + 10 a + 52\right)\cdot 83^{7} + \left(65 a^{2} + 6 a + 60\right)\cdot 83^{8} + \left(69 a^{2} + 38 a + 63\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 81 a^{2} + 27 a + 4 + \left(2 a^{2} + 32 a + 66\right)\cdot 83 + \left(32 a^{2} + 13 a + 57\right)\cdot 83^{2} + \left(80 a^{2} + 79 a + 56\right)\cdot 83^{3} + \left(72 a^{2} + 12 a + 79\right)\cdot 83^{4} + \left(53 a^{2} + 77 a + 55\right)\cdot 83^{5} + \left(26 a^{2} + 25\right)\cdot 83^{6} + \left(32 a^{2} + 31 a + 10\right)\cdot 83^{7} + \left(80 a^{2} + 7\right)\cdot 83^{8} + \left(23 a^{2} + 73 a + 55\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a^{2} + 75 a + 10 + \left(40 a^{2} + 3 a + 24\right)\cdot 83 + \left(33 a + 27\right)\cdot 83^{2} + \left(25 a^{2} + 38 a + 69\right)\cdot 83^{3} + \left(38 a^{2} + 65 a + 30\right)\cdot 83^{4} + \left(74 a^{2} + 82 a + 1\right)\cdot 83^{5} + \left(70 a^{2} + 75 a + 57\right)\cdot 83^{6} + \left(80 a^{2} + a + 12\right)\cdot 83^{7} + \left(77 a^{2} + 10 a + 14\right)\cdot 83^{8} + \left(76 a^{2} + 37 a + 23\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 73 a^{2} + 38 a + 49 + \left(76 a^{2} + 53 a + 14\right)\cdot 83 + \left(37 a + 28\right)\cdot 83^{2} + \left(32 a^{2} + 44 a\right)\cdot 83^{3} + \left(58 a^{2} + a + 71\right)\cdot 83^{4} + \left(58 a^{2} + 24 a + 52\right)\cdot 83^{5} + \left(23 a^{2} + 74 a + 45\right)\cdot 83^{6} + \left(50 a^{2} + 27 a + 34\right)\cdot 83^{7} + \left(46 a^{2} + 2 a + 34\right)\cdot 83^{8} + \left(71 a^{2} + 10 a + 12\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 40 a^{2} + 33 a + 5 + \left(66 a^{2} + 79 a + 27\right)\cdot 83 + \left(47 a^{2} + a + 6\right)\cdot 83^{2} + \left(18 a^{2} + 39 a + 16\right)\cdot 83^{3} + \left(25 a^{2} + 55 a + 67\right)\cdot 83^{4} + \left(62 a^{2} + 42 a + 72\right)\cdot 83^{5} + \left(9 a^{2} + 41 a + 74\right)\cdot 83^{6} + \left(80 a^{2} + 41 a + 22\right)\cdot 83^{7} + \left(19 a^{2} + 76 a + 52\right)\cdot 83^{8} + \left(72 a^{2} + 54 a + 68\right)\cdot 83^{9} +O(83^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(3,5)(4,8)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $3$ |
| $24$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $36$ | $2$ | $(1,2)(3,5)(4,8)$ | $-1$ |
| $16$ | $3$ | $(1,6,7)$ | $0$ |
| $64$ | $3$ | $(1,6,7)(4,5,8)$ | $0$ |
| $12$ | $4$ | $(3,4,5,8)$ | $-3$ |
| $36$ | $4$ | $(1,2,6,7)(3,4,5,8)$ | $1$ |
| $36$ | $4$ | $(1,2,6,7)(3,5)(4,8)$ | $1$ |
| $72$ | $4$ | $(1,3,6,5)(2,4,7,8)$ | $-1$ |
| $72$ | $4$ | $(1,2)(3,4,5,8)$ | $-1$ |
| $144$ | $4$ | $(1,4,2,3)(5,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,7,6)(3,5)(4,8)$ | $0$ |
| $96$ | $6$ | $(1,2)(4,8,5)$ | $0$ |
| $192$ | $6$ | $(1,4,6,5,7,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,3,2,4,6,5,7,8)$ | $-1$ |
| $96$ | $12$ | $(1,6,7)(3,4,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.