Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(522324376576\)\(\medspace = 2^{10} \cdot 17^{3} \cdot 47^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1004924672.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.799.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.1004924672.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} - 4x^{5} + 7x^{4} + 8x^{3} - 2x^{2} - 6x - 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{3} + 3x + 51 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a^{2} + 11 a + 15 + \left(44 a^{2} + 8 a + 52\right)\cdot 53 + \left(29 a^{2} + 31 a + 6\right)\cdot 53^{2} + \left(12 a^{2} + 9 a + 32\right)\cdot 53^{3} + \left(47 a^{2} + 25 a + 34\right)\cdot 53^{4} + \left(31 a^{2} + 18 a + 20\right)\cdot 53^{5} + \left(45 a^{2} + 47 a + 18\right)\cdot 53^{6} + \left(33 a^{2} + 33 a + 4\right)\cdot 53^{7} + \left(45 a^{2} + 49 a + 31\right)\cdot 53^{8} + \left(15 a^{2} + 12 a + 4\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a^{2} + 33 a + 3 + \left(39 a^{2} + 25 a + 42\right)\cdot 53 + \left(43 a^{2} + 23 a + 34\right)\cdot 53^{2} + \left(3 a^{2} + 14\right)\cdot 53^{3} + \left(12 a + 46\right)\cdot 53^{4} + \left(51 a^{2} + 36 a + 5\right)\cdot 53^{5} + \left(45 a^{2} + 13 a + 19\right)\cdot 53^{6} + \left(17 a^{2} + 43 a + 25\right)\cdot 53^{7} + \left(34 a^{2} + 27 a + 8\right)\cdot 53^{8} + \left(16 a^{2} + 9 a + 6\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a^{2} + 46 a + 14 + \left(23 a^{2} + 25 a + 51\right)\cdot 53 + \left(25 a^{2} + 52 a + 45\right)\cdot 53^{2} + \left(25 a^{2} + 17 a + 15\right)\cdot 53^{3} + \left(34 a^{2} + 30 a + 25\right)\cdot 53^{4} + \left(4 a^{2} + 38 a + 21\right)\cdot 53^{5} + \left(23 a^{2} + 51 a + 47\right)\cdot 53^{6} + \left(43 a^{2} + 11 a + 20\right)\cdot 53^{7} + \left(51 a^{2} + 32 a + 5\right)\cdot 53^{8} + \left(18 a^{2} + 22 a + 31\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 a^{2} + 9 a + 43 + \left(22 a^{2} + 19 a + 8\right)\cdot 53 + \left(32 a^{2} + 51 a + 12\right)\cdot 53^{2} + \left(36 a^{2} + 42 a + 27\right)\cdot 53^{3} + \left(5 a^{2} + 15 a + 4\right)\cdot 53^{4} + \left(23 a^{2} + 51 a + 3\right)\cdot 53^{5} + \left(14 a^{2} + 44 a + 9\right)\cdot 53^{6} + \left(a^{2} + 28 a + 45\right)\cdot 53^{7} + \left(26 a^{2} + 28 a + 44\right)\cdot 53^{8} + \left(20 a^{2} + 30 a + 13\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 2\cdot 53 + 52\cdot 53^{2} + 31\cdot 53^{3} + 20\cdot 53^{4} + 23\cdot 53^{5} + 6\cdot 53^{6} + 31\cdot 53^{7} + 21\cdot 53^{8} + 28\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a^{2} + 49 a + 16 + \left(11 a^{2} + 4 a + 27\right)\cdot 53 + \left(19 a^{2} + 33 a + 33\right)\cdot 53^{2} + \left(29 a^{2} + 40 a + 23\right)\cdot 53^{3} + \left(13 a^{2} + 41 a + 36\right)\cdot 53^{4} + \left(22 a^{2} + 28 a + 3\right)\cdot 53^{5} + \left(13 a^{2} + 28\right)\cdot 53^{6} + \left(32 a^{2} + 51\right)\cdot 53^{7} + \left(32 a^{2} + 23 a + 19\right)\cdot 53^{8} + \left(46 a^{2} + 15 a + 33\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 18 + 38\cdot 53 + 14\cdot 53^{2} + 52\cdot 53^{3} + 24\cdot 53^{4} + 16\cdot 53^{5} + 49\cdot 53^{6} + 38\cdot 53^{7} + 29\cdot 53^{8} + 20\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 32 a^{2} + 11 a + 5 + \left(18 a^{2} + 22 a + 42\right)\cdot 53 + \left(8 a^{2} + 20 a + 11\right)\cdot 53^{2} + \left(51 a^{2} + 47 a + 14\right)\cdot 53^{3} + \left(4 a^{2} + 33 a + 19\right)\cdot 53^{4} + \left(26 a^{2} + 38 a + 11\right)\cdot 53^{5} + \left(16 a^{2} + 34\right)\cdot 53^{6} + \left(30 a^{2} + 41 a + 47\right)\cdot 53^{7} + \left(21 a^{2} + 50 a + 50\right)\cdot 53^{8} + \left(40 a^{2} + 14 a + 20\right)\cdot 53^{9} +O(53^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(3,7)(6,8)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $3$ |
| $24$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(3,8)$ | $1$ |
| $36$ | $2$ | $(1,2)(3,7)(6,8)$ | $-1$ |
| $16$ | $3$ | $(2,5,4)$ | $0$ |
| $64$ | $3$ | $(2,5,4)(6,8,7)$ | $0$ |
| $12$ | $4$ | $(3,6,7,8)$ | $-3$ |
| $36$ | $4$ | $(1,2,4,5)(3,6,7,8)$ | $1$ |
| $36$ | $4$ | $(1,2,4,5)(3,7)(6,8)$ | $1$ |
| $72$ | $4$ | $(1,3,4,7)(2,6,5,8)$ | $-1$ |
| $72$ | $4$ | $(1,2)(3,6,7,8)$ | $-1$ |
| $144$ | $4$ | $(1,3,2,8)(4,6)(5,7)$ | $1$ |
| $48$ | $6$ | $(2,4,5)(3,7)(6,8)$ | $0$ |
| $96$ | $6$ | $(2,5,4)(3,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,8,5,7,4,6)$ | $0$ |
| $144$ | $8$ | $(1,3,2,6,4,7,5,8)$ | $-1$ |
| $96$ | $12$ | $(2,5,4)(3,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.