Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(120846864384\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 103^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1383025225728.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.1236.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.1383025225728.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} + 6x^{5} - 47x^{4} + 94x^{3} - 82x^{2} + 26x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{3} + x + 40 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a^{2} + 33 a + 32 + \left(17 a^{2} + 28 a + 26\right)\cdot 43 + \left(33 a^{2} + 18 a + 27\right)\cdot 43^{2} + \left(18 a^{2} + 35 a + 30\right)\cdot 43^{3} + \left(20 a^{2} + 33 a + 40\right)\cdot 43^{4} + \left(16 a^{2} + 6 a + 21\right)\cdot 43^{5} + \left(27 a^{2} + 14 a + 25\right)\cdot 43^{6} + \left(14 a^{2} + 18 a + 18\right)\cdot 43^{7} + \left(a^{2} + 41 a + 10\right)\cdot 43^{8} + \left(38 a^{2} + 27 a + 36\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 40\cdot 43 + 26\cdot 43^{2} + 31\cdot 43^{3} + 4\cdot 43^{4} + 10\cdot 43^{5} + 21\cdot 43^{6} + 16\cdot 43^{7} + 14\cdot 43^{8} + 10\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 a^{2} + 33 a + 42 + \left(42 a^{2} + 27 a + 14\right)\cdot 43 + \left(15 a^{2} + 7 a + 30\right)\cdot 43^{2} + \left(36 a^{2} + 20 a + 13\right)\cdot 43^{3} + \left(23 a^{2} + 26 a + 14\right)\cdot 43^{4} + \left(17 a^{2} + 12 a + 8\right)\cdot 43^{5} + \left(7 a^{2} + 41 a + 12\right)\cdot 43^{6} + \left(29 a^{2} + 28\right)\cdot 43^{7} + \left(39 a^{2} + 10 a + 21\right)\cdot 43^{8} + \left(11 a^{2} + 41 a + 4\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 a^{2} + 9 a + 34 + \left(38 a^{2} + 31 a + 40\right)\cdot 43 + \left(38 a^{2} + 5 a + 39\right)\cdot 43^{2} + \left(14 a + 17\right)\cdot 43^{3} + \left(3 a^{2} + 19 a + 40\right)\cdot 43^{4} + \left(15 a^{2} + 13 a + 37\right)\cdot 43^{5} + \left(37 a^{2} + 15 a + 40\right)\cdot 43^{6} + \left(17 a^{2} + 17 a + 10\right)\cdot 43^{7} + \left(7 a^{2} + 17 a + 35\right)\cdot 43^{8} + \left(15 a^{2} + 29 a + 20\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a^{2} + 29 a + 7 + \left(6 a^{2} + 40 a + 19\right)\cdot 43 + \left(20 a^{2} + 13 a + 27\right)\cdot 43^{2} + \left(6 a^{2} + 24 a + 21\right)\cdot 43^{3} + \left(5 a^{2} + 33 a + 27\right)\cdot 43^{4} + \left(12 a^{2} + 31 a + 21\right)\cdot 43^{5} + \left(41 a^{2} + 25 a\right)\cdot 43^{6} + \left(42 a^{2} + 22 a + 42\right)\cdot 43^{7} + \left(28 a^{2} + 41 a + 20\right)\cdot 43^{8} + \left(11 a^{2} + 40 a + 18\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 + 41\cdot 43 + 34\cdot 43^{3} + 13\cdot 43^{4} + 2\cdot 43^{5} + 38\cdot 43^{6} + 2\cdot 43^{7} + 38\cdot 43^{8} + 10\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 35 a^{2} + 5 a + 34 + \left(40 a^{2} + 14 a + 27\right)\cdot 43 + \left(26 a^{2} + 23 a + 17\right)\cdot 43^{2} + \left(35 a^{2} + 4 a + 12\right)\cdot 43^{3} + \left(34 a^{2} + 33 a + 4\right)\cdot 43^{4} + \left(15 a^{2} + 40 a + 24\right)\cdot 43^{5} + \left(7 a^{2} + a + 6\right)\cdot 43^{6} + \left(25 a^{2} + 3 a + 30\right)\cdot 43^{7} + \left(6 a^{2} + 27 a + 34\right)\cdot 43^{8} + \left(16 a^{2} + 15 a + 35\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 31 a^{2} + 20 a + 25 + \left(25 a^{2} + 29 a + 3\right)\cdot 43 + \left(36 a^{2} + 16 a + 1\right)\cdot 43^{2} + \left(30 a^{2} + 30 a + 10\right)\cdot 43^{3} + \left(41 a^{2} + 25 a + 26\right)\cdot 43^{4} + \left(8 a^{2} + 23 a + 2\right)\cdot 43^{5} + \left(8 a^{2} + 30 a + 27\right)\cdot 43^{6} + \left(42 a^{2} + 23 a + 22\right)\cdot 43^{7} + \left(a^{2} + 34 a + 39\right)\cdot 43^{8} + \left(36 a^{2} + 16 a + 34\right)\cdot 43^{9} +O(43^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,3)(2,8)$ | $-3$ |
| $9$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $3$ |
| $24$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $36$ | $2$ | $(1,2)(4,6)(5,7)$ | $-1$ |
| $16$ | $3$ | $(1,3,8)$ | $0$ |
| $64$ | $3$ | $(1,3,8)(4,6,7)$ | $0$ |
| $12$ | $4$ | $(1,2,3,8)$ | $-3$ |
| $36$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | $1$ |
| $36$ | $4$ | $(1,3)(2,8)(4,5,6,7)$ | $1$ |
| $72$ | $4$ | $(1,6,3,4)(2,7,8,5)$ | $-1$ |
| $72$ | $4$ | $(1,2)(4,5,6,7)$ | $-1$ |
| $144$ | $4$ | $(1,5,2,4)(3,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,8,3)(4,6)(5,7)$ | $0$ |
| $96$ | $6$ | $(1,3,8)(4,5)$ | $0$ |
| $192$ | $6$ | $(1,6,3,7,8,4)(2,5)$ | $0$ |
| $144$ | $8$ | $(1,5,2,6,3,7,8,4)$ | $-1$ |
| $96$ | $12$ | $(1,3,8)(4,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.