Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(877897859072\)\(\medspace = 2^{14} \cdot 13^{3} \cdot 29^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1292841769024.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.1508.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.1292841769024.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} - 38x^{5} + 25x^{4} - 76x^{3} + 26x^{2} - 22x + 16 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 2 + \left(21 a + 12\right)\cdot 47 + \left(3 a + 16\right)\cdot 47^{2} + \left(13 a + 31\right)\cdot 47^{3} + \left(45 a + 20\right)\cdot 47^{4} + \left(35 a + 38\right)\cdot 47^{5} + \left(39 a + 36\right)\cdot 47^{6} + \left(12 a + 8\right)\cdot 47^{7} + \left(41 a + 8\right)\cdot 47^{8} + \left(43 a + 10\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 40\cdot 47 + 25\cdot 47^{2} + 20\cdot 47^{3} + 34\cdot 47^{4} + 6\cdot 47^{5} + 42\cdot 47^{6} + 27\cdot 47^{7} + 27\cdot 47^{8} + 29\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 30 + \left(25 a + 40\right)\cdot 47 + \left(43 a + 1\right)\cdot 47^{2} + \left(33 a + 7\right)\cdot 47^{3} + \left(a + 4\right)\cdot 47^{4} + \left(11 a + 18\right)\cdot 47^{5} + \left(7 a + 33\right)\cdot 47^{6} + \left(34 a + 41\right)\cdot 47^{7} + \left(5 a + 30\right)\cdot 47^{8} + \left(3 a + 9\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 19 + \left(41 a + 32\right)\cdot 47 + \left(40 a + 17\right)\cdot 47^{2} + \left(29 a + 18\right)\cdot 47^{3} + \left(41 a + 31\right)\cdot 47^{4} + \left(5 a + 33\right)\cdot 47^{5} + \left(37 a + 24\right)\cdot 47^{6} + \left(34 a + 5\right)\cdot 47^{7} + \left(19 a + 25\right)\cdot 47^{8} + \left(38 a + 8\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 + 16\cdot 47 + 32\cdot 47^{2} + 44\cdot 47^{3} + 25\cdot 47^{4} + 21\cdot 47^{5} + 19\cdot 47^{6} + 46\cdot 47^{7} + 34\cdot 47^{8} + 46\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a + 43 + \left(5 a + 8\right)\cdot 47 + \left(6 a + 11\right)\cdot 47^{2} + \left(17 a + 37\right)\cdot 47^{3} + \left(5 a + 37\right)\cdot 47^{4} + \left(41 a + 3\right)\cdot 47^{5} + \left(9 a + 46\right)\cdot 47^{6} + \left(12 a + 37\right)\cdot 47^{7} + \left(27 a + 29\right)\cdot 47^{8} + \left(8 a + 18\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 a + \left(42 a + 39\right)\cdot 47 + \left(9 a + 5\right)\cdot 47^{2} + \left(22 a + 44\right)\cdot 47^{3} + \left(43 a + 7\right)\cdot 47^{4} + \left(44 a + 33\right)\cdot 47^{5} + \left(43 a + 41\right)\cdot 47^{6} + \left(24 a + 6\right)\cdot 47^{7} + \left(8 a + 43\right)\cdot 47^{8} + \left(32 a + 27\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 15 a + 17 + \left(4 a + 45\right)\cdot 47 + \left(37 a + 29\right)\cdot 47^{2} + \left(24 a + 31\right)\cdot 47^{3} + \left(3 a + 25\right)\cdot 47^{4} + \left(2 a + 32\right)\cdot 47^{5} + \left(3 a + 37\right)\cdot 47^{6} + \left(22 a + 12\right)\cdot 47^{7} + \left(38 a + 35\right)\cdot 47^{8} + \left(14 a + 36\right)\cdot 47^{9} +O(47^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,7)(5,8)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $1$ |
| $12$ | $2$ | $(1,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $3$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $1$ |
| $36$ | $2$ | $(1,3)(2,7)(5,8)$ | $-1$ |
| $16$ | $3$ | $(1,4,6)$ | $0$ |
| $64$ | $3$ | $(1,4,6)(5,7,8)$ | $0$ |
| $12$ | $4$ | $(2,5,7,8)$ | $-3$ |
| $36$ | $4$ | $(1,3,4,6)(2,5,7,8)$ | $1$ |
| $36$ | $4$ | $(1,3,4,6)(2,7)(5,8)$ | $1$ |
| $72$ | $4$ | $(1,2,4,7)(3,5,6,8)$ | $-1$ |
| $72$ | $4$ | $(1,3)(2,5,7,8)$ | $-1$ |
| $144$ | $4$ | $(1,5,3,2)(4,7)(6,8)$ | $1$ |
| $48$ | $6$ | $(1,6,4)(2,7)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(5,8,7)$ | $0$ |
| $192$ | $6$ | $(1,5,4,7,6,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,5,4,7,6,8)$ | $-1$ |
| $96$ | $12$ | $(1,4,6)(2,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.