Basic invariants
| Dimension: | $6$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(7972272736871875\)\(\medspace = 5^{5} \cdot 19^{5} \cdot 101^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.9885738095.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T202 |
| Parity: | odd |
| Determinant: | 1.9595.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.9885738095.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 5x^{6} - 4x^{5} - x^{4} + 5x^{3} - 10x^{2} - 19 \)
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The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 25 + \left(13 a + 12\right)\cdot 43 + \left(26 a + 30\right)\cdot 43^{2} + \left(18 a + 15\right)\cdot 43^{3} + \left(37 a + 19\right)\cdot 43^{4} + \left(37 a + 1\right)\cdot 43^{5} + \left(37 a + 20\right)\cdot 43^{6} + \left(22 a + 24\right)\cdot 43^{7} + \left(2 a + 27\right)\cdot 43^{8} + \left(20 a + 17\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 10 + \left(17 a + 28\right)\cdot 43 + \left(20 a + 20\right)\cdot 43^{2} + \left(25 a + 5\right)\cdot 43^{3} + \left(5 a + 24\right)\cdot 43^{4} + \left(25 a + 15\right)\cdot 43^{5} + \left(30 a + 41\right)\cdot 43^{6} + \left(15 a + 12\right)\cdot 43^{7} + \left(20 a + 40\right)\cdot 43^{8} + \left(2 a + 36\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 26 + \left(2 a + 12\right)\cdot 43 + \left(10 a + 38\right)\cdot 43^{2} + \left(6 a + 36\right)\cdot 43^{3} + \left(16 a + 23\right)\cdot 43^{4} + \left(9 a + 42\right)\cdot 43^{5} + \left(3 a + 1\right)\cdot 43^{6} + \left(8 a + 35\right)\cdot 43^{7} + \left(2 a + 24\right)\cdot 43^{8} + \left(20 a + 27\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 17\cdot 43 + 30\cdot 43^{2} + 30\cdot 43^{3} + 27\cdot 43^{4} + 27\cdot 43^{5} + 12\cdot 43^{6} + 21\cdot 43^{7} + 29\cdot 43^{8} + 36\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 31\cdot 43 + 24\cdot 43^{2} + 31\cdot 43^{3} + 12\cdot 43^{5} + 33\cdot 43^{6} + 30\cdot 43^{7} + 21\cdot 43^{8} + 39\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 a + 26 + \left(29 a + 24\right)\cdot 43 + 16 a\cdot 43^{2} + \left(24 a + 8\right)\cdot 43^{3} + \left(5 a + 38\right)\cdot 43^{4} + \left(5 a + 1\right)\cdot 43^{5} + \left(5 a + 20\right)\cdot 43^{6} + \left(20 a + 9\right)\cdot 43^{7} + \left(40 a + 7\right)\cdot 43^{8} + \left(22 a + 35\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 a + 19 + \left(25 a + 36\right)\cdot 43 + \left(22 a + 23\right)\cdot 43^{2} + \left(17 a + 10\right)\cdot 43^{3} + \left(37 a + 4\right)\cdot 43^{4} + \left(17 a + 35\right)\cdot 43^{5} + \left(12 a + 3\right)\cdot 43^{6} + \left(27 a + 41\right)\cdot 43^{7} + \left(22 a + 1\right)\cdot 43^{8} + \left(40 a + 19\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 37 a + 32 + \left(40 a + 8\right)\cdot 43 + \left(32 a + 3\right)\cdot 43^{2} + \left(36 a + 33\right)\cdot 43^{3} + \left(26 a + 33\right)\cdot 43^{4} + \left(33 a + 35\right)\cdot 43^{5} + \left(39 a + 38\right)\cdot 43^{6} + \left(34 a + 39\right)\cdot 43^{7} + \left(40 a + 18\right)\cdot 43^{8} + \left(22 a + 2\right)\cdot 43^{9} +O(43^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $6$ | $2$ | $(1,5)(4,6)$ | $2$ |
| $9$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $-2$ |
| $12$ | $2$ | $(2,3)$ | $-4$ |
| $24$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $2$ |
| $36$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $16$ | $3$ | $(2,7,8)$ | $3$ |
| $64$ | $3$ | $(2,7,8)(4,5,6)$ | $0$ |
| $12$ | $4$ | $(1,4,5,6)$ | $-2$ |
| $36$ | $4$ | $(1,4,5,6)(2,3,7,8)$ | $-2$ |
| $36$ | $4$ | $(1,5)(2,3,7,8)(4,6)$ | $2$ |
| $72$ | $4$ | $(1,7,5,2)(3,4,8,6)$ | $0$ |
| $72$ | $4$ | $(1,4,5,6)(2,3)$ | $0$ |
| $144$ | $4$ | $(1,2,4,3)(5,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(1,5)(2,8,7)(4,6)$ | $-1$ |
| $96$ | $6$ | $(2,3)(4,6,5)$ | $-1$ |
| $192$ | $6$ | $(1,3)(2,4,7,5,8,6)$ | $0$ |
| $144$ | $8$ | $(1,3,4,7,5,8,6,2)$ | $0$ |
| $96$ | $12$ | $(1,4,5,6)(2,7,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.