Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(887423028741\)\(\medspace = 3^{4} \cdot 2221^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.6.656988848944587.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T272 |
| Parity: | even |
| Determinant: | 1.2221.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.6.656988848944587.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 25x^{6} + 87x^{5} + 241x^{4} - 926x^{3} - 1354x^{2} + 2946x + 3867 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{3} + x + 28 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a^{2} + 14 a + 27 + \left(23 a^{2} + 19 a + 11\right)\cdot 31 + \left(12 a^{2} + 4 a + 20\right)\cdot 31^{2} + \left(21 a^{2} + 5 a + 7\right)\cdot 31^{3} + \left(28 a^{2} + 21 a + 23\right)\cdot 31^{4} + \left(5 a^{2} + 14 a + 28\right)\cdot 31^{5} + \left(a^{2} + 22 a + 30\right)\cdot 31^{6} + \left(24 a^{2} + 10 a + 30\right)\cdot 31^{7} + \left(14 a^{2} + 11 a + 1\right)\cdot 31^{8} + \left(28 a^{2} + 10 a + 4\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 a^{2} + 18 a + 22 + \left(25 a^{2} + 28 a + 14\right)\cdot 31 + \left(4 a^{2} + 30 a + 12\right)\cdot 31^{2} + \left(15 a^{2} + 11 a + 12\right)\cdot 31^{3} + \left(15 a^{2} + 8 a + 13\right)\cdot 31^{4} + \left(25 a^{2} + 30 a + 3\right)\cdot 31^{5} + \left(21 a^{2} + a + 11\right)\cdot 31^{6} + \left(14 a^{2} + 12 a + 1\right)\cdot 31^{7} + \left(2 a^{2} + 23 a + 20\right)\cdot 31^{8} + \left(4 a^{2} + 24 a + 9\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a^{2} + 3 a + 19 + \left(4 a^{2} + 20 a + 19\right)\cdot 31 + \left(23 a^{2} + 27 a + 6\right)\cdot 31^{2} + \left(8 a^{2} + 13 a + 30\right)\cdot 31^{3} + \left(15 a^{2} + 24 a + 3\right)\cdot 31^{4} + \left(28 a^{2} + 23 a + 23\right)\cdot 31^{5} + \left(8 a^{2} + 17 a + 25\right)\cdot 31^{6} + \left(13 a^{2} + 23\right)\cdot 31^{7} + \left(23 a^{2} + 23 a + 7\right)\cdot 31^{8} + \left(13 a^{2} + 25 a + 25\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a^{2} + 5 a + 7 + \left(7 a^{2} + 3 a + 23\right)\cdot 31 + \left(14 a^{2} + 18 a + 18\right)\cdot 31^{2} + \left(24 a^{2} + 4 a + 18\right)\cdot 31^{3} + \left(8 a^{2} + 15 a + 29\right)\cdot 31^{4} + \left(4 a^{2} + 11 a + 9\right)\cdot 31^{5} + \left(16 a^{2} + 25 a + 7\right)\cdot 31^{6} + \left(8 a^{2} + 20 a + 28\right)\cdot 31^{7} + \left(a^{2} + 27 a + 8\right)\cdot 31^{8} + \left(12 a^{2} + 27 a + 25\right)\cdot 31^{9} +O(31^{10})\)
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| $r_{ 5 }$ | $=$ |
\( 25 a^{2} + 8 a + 24 + \left(28 a^{2} + 30 a + 16\right)\cdot 31 + \left(11 a^{2} + 12 a + 27\right)\cdot 31^{2} + \left(22 a^{2} + 14 a + 6\right)\cdot 31^{3} + \left(6 a^{2} + 7 a + 28\right)\cdot 31^{4} + \left(a^{2} + 20 a + 7\right)\cdot 31^{5} + \left(24 a^{2} + 3 a + 2\right)\cdot 31^{6} + \left(7 a^{2} + 29 a + 7\right)\cdot 31^{7} + \left(27 a^{2} + 10 a + 26\right)\cdot 31^{8} + \left(14 a^{2} + 9 a + 16\right)\cdot 31^{9} +O(31^{10})\)
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| $r_{ 6 }$ | $=$ |
\( 10 + 7\cdot 31 + 3\cdot 31^{2} + 24\cdot 31^{3} + 21\cdot 31^{4} + 9\cdot 31^{5} + 10\cdot 31^{6} + 25\cdot 31^{7} + 6\cdot 31^{8} + 10\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 a^{2} + 14 a + 30 + \left(3 a^{2} + 22 a + 18\right)\cdot 31 + \left(26 a^{2} + 29 a + 8\right)\cdot 31^{2} + \left(11 a + 4\right)\cdot 31^{3} + \left(18 a^{2} + 16 a + 16\right)\cdot 31^{4} + \left(27 a^{2} + 23 a + 22\right)\cdot 31^{5} + \left(20 a^{2} + 21 a + 2\right)\cdot 31^{6} + \left(24 a^{2} + 19 a + 21\right)\cdot 31^{7} + \left(23 a^{2} + 27 a + 28\right)\cdot 31^{8} + \left(19 a^{2} + 25 a + 18\right)\cdot 31^{9} +O(31^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 18 + 11\cdot 31 + 26\cdot 31^{2} + 19\cdot 31^{3} + 18\cdot 31^{4} + 18\cdot 31^{5} + 2\cdot 31^{6} + 17\cdot 31^{7} + 23\cdot 31^{8} + 13\cdot 31^{9} +O(31^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,5)(4,6)$ | $-3$ |
| $9$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $1$ |
| $12$ | $2$ | $(1,3)$ | $-3$ |
| $24$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $3$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $1$ |
| $36$ | $2$ | $(1,3)(2,5)(4,6)$ | $1$ |
| $16$ | $3$ | $(1,7,8)$ | $0$ |
| $64$ | $3$ | $(1,7,8)(4,5,6)$ | $0$ |
| $12$ | $4$ | $(2,4,5,6)$ | $3$ |
| $36$ | $4$ | $(1,3,7,8)(2,4,5,6)$ | $1$ |
| $36$ | $4$ | $(1,3,7,8)(2,5)(4,6)$ | $-1$ |
| $72$ | $4$ | $(1,2,7,5)(3,4,8,6)$ | $-1$ |
| $72$ | $4$ | $(1,3)(2,4,5,6)$ | $-1$ |
| $144$ | $4$ | $(1,4,3,2)(5,7)(6,8)$ | $-1$ |
| $48$ | $6$ | $(1,8,7)(2,5)(4,6)$ | $0$ |
| $96$ | $6$ | $(1,3)(4,6,5)$ | $0$ |
| $192$ | $6$ | $(1,4,7,5,8,6)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,4,7,5,8,6)$ | $1$ |
| $96$ | $12$ | $(1,7,8)(2,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.