Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr S_4$ |
| Conductor: | \(1538820023\)\(\medspace = 13 \cdot 491^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.3134053.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr S_4$ |
| Parity: | odd |
| Determinant: | 1.6383.2t1.a.a |
| Projective image: | $C_2^3:S_4$ |
| Projective stem field: | Galois closure of 8.4.6885514441.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + x^{6} - 2x^{5} + 3x^{4} - 2x^{3} + x^{2} - x + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 29 a + 42 + \left(38 a + 8\right)\cdot 47 + \left(14 a + 2\right)\cdot 47^{2} + \left(28 a + 17\right)\cdot 47^{3} + \left(12 a + 33\right)\cdot 47^{4} + \left(2 a + 17\right)\cdot 47^{5} + \left(5 a + 34\right)\cdot 47^{6} + \left(24 a + 43\right)\cdot 47^{7} + \left(23 a + 28\right)\cdot 47^{8} + 37\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 + 44\cdot 47 + 22\cdot 47^{2} + 8\cdot 47^{3} + 6\cdot 47^{4} + 31\cdot 47^{5} + 30\cdot 47^{6} + 6\cdot 47^{7} + 9\cdot 47^{8} + 15\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 6 + \left(8 a + 10\right)\cdot 47 + \left(32 a + 40\right)\cdot 47^{2} + \left(18 a + 11\right)\cdot 47^{3} + \left(34 a + 30\right)\cdot 47^{4} + \left(44 a + 9\right)\cdot 47^{5} + \left(41 a + 42\right)\cdot 47^{6} + \left(22 a + 39\right)\cdot 47^{7} + \left(23 a + 4\right)\cdot 47^{8} + \left(46 a + 15\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 1 + \left(41 a + 37\right)\cdot 47 + \left(31 a + 44\right)\cdot 47^{2} + \left(42 a + 20\right)\cdot 47^{3} + \left(32 a + 8\right)\cdot 47^{4} + \left(23 a + 33\right)\cdot 47^{5} + \left(46 a + 43\right)\cdot 47^{6} + \left(41 a + 9\right)\cdot 47^{7} + \left(21 a + 46\right)\cdot 47^{8} + \left(46 a + 34\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 a + 19 + \left(5 a + 16\right)\cdot 47 + \left(15 a + 20\right)\cdot 47^{2} + \left(4 a + 27\right)\cdot 47^{3} + \left(14 a + 31\right)\cdot 47^{4} + 23 a\cdot 47^{5} + 19\cdot 47^{6} + 5 a\cdot 47^{7} + \left(25 a + 1\right)\cdot 47^{8} + 12\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 30 + \left(14 a + 30\right)\cdot 47 + \left(36 a + 33\right)\cdot 47^{2} + \left(15 a + 41\right)\cdot 47^{3} + \left(29 a + 2\right)\cdot 47^{4} + \left(40 a + 15\right)\cdot 47^{5} + \left(42 a + 11\right)\cdot 47^{6} + \left(20 a + 30\right)\cdot 47^{7} + \left(45 a + 39\right)\cdot 47^{8} + \left(19 a + 33\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 41\cdot 47 + 25\cdot 47^{2} + 23\cdot 47^{3} + 29\cdot 47^{4} + 13\cdot 47^{5} + 44\cdot 47^{6} + 27\cdot 47^{7} + 42\cdot 47^{8} + 10\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 33 a + 11 + \left(32 a + 45\right)\cdot 47 + \left(10 a + 44\right)\cdot 47^{2} + \left(31 a + 36\right)\cdot 47^{3} + \left(17 a + 45\right)\cdot 47^{4} + \left(6 a + 19\right)\cdot 47^{5} + \left(4 a + 9\right)\cdot 47^{6} + \left(26 a + 29\right)\cdot 47^{7} + \left(a + 15\right)\cdot 47^{8} + \left(27 a + 28\right)\cdot 47^{9} +O(47^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $4$ | $2$ | $(1,8)$ | $2$ |
| $4$ | $2$ | $(1,8)(2,7)(4,5)$ | $-2$ |
| $6$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(3,4)(5,6)$ | $-2$ |
| $12$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $2$ |
| $24$ | $2$ | $(1,8)(3,4)(5,6)$ | $0$ |
| $32$ | $3$ | $(1,2,3)(6,8,7)$ | $1$ |
| $12$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $12$ | $4$ | $(3,4,6,5)$ | $-2$ |
| $12$ | $4$ | $(1,7,8,2)(3,6)(4,5)$ | $2$ |
| $24$ | $4$ | $(1,6,8,3)(2,4)(5,7)$ | $0$ |
| $24$ | $4$ | $(1,8)(3,4,6,5)$ | $0$ |
| $48$ | $4$ | $(1,2,3,4)(5,8,7,6)$ | $0$ |
| $32$ | $6$ | $(1,5,7,8,4,2)$ | $1$ |
| $32$ | $6$ | $(1,2,3)(4,5)(6,8,7)$ | $-1$ |
| $32$ | $6$ | $(1,7,6,8,2,3)(4,5)$ | $-1$ |
| $48$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.