Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr S_4$ |
| Conductor: | \(5871\)\(\medspace = 3 \cdot 19 \cdot 103 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.11489547.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr S_4$ |
| Parity: | odd |
| Determinant: | 1.5871.2t1.a.a |
| Projective image: | $C_2^3:S_4$ |
| Projective stem field: | Galois closure of 8.0.310217769.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{5} - x^{4} - x^{3} + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 40 a + 6 + \left(46 a + 6\right)\cdot 47 + \left(43 a + 12\right)\cdot 47^{2} + \left(3 a + 20\right)\cdot 47^{3} + \left(27 a + 16\right)\cdot 47^{4} + \left(36 a + 42\right)\cdot 47^{5} + \left(42 a + 20\right)\cdot 47^{6} + \left(29 a + 40\right)\cdot 47^{7} + \left(41 a + 17\right)\cdot 47^{8} + \left(26 a + 1\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 24 + \left(5 a + 3\right)\cdot 47 + \left(31 a + 32\right)\cdot 47^{2} + \left(a + 6\right)\cdot 47^{3} + \left(4 a + 8\right)\cdot 47^{4} + \left(43 a + 7\right)\cdot 47^{5} + \left(11 a + 5\right)\cdot 47^{6} + \left(3 a + 22\right)\cdot 47^{7} + \left(a + 32\right)\cdot 47^{8} + \left(13 a + 38\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 + 34\cdot 47 + 34\cdot 47^{2} + 13\cdot 47^{3} + 14\cdot 47^{4} + 39\cdot 47^{5} + 16\cdot 47^{6} + 6\cdot 47^{7} + 37\cdot 47^{8} + 36\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 a + 32 + \left(27 a + 41\right)\cdot 47 + \left(4 a + 35\right)\cdot 47^{2} + \left(45 a + 7\right)\cdot 47^{3} + \left(16 a + 45\right)\cdot 47^{4} + \left(2 a + 39\right)\cdot 47^{5} + \left(43 a + 44\right)\cdot 47^{6} + \left(43 a + 40\right)\cdot 47^{7} + \left(41 a + 30\right)\cdot 47^{8} + \left(a + 38\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a + 39 + 12\cdot 47 + \left(3 a + 6\right)\cdot 47^{2} + \left(43 a + 31\right)\cdot 47^{3} + \left(19 a + 19\right)\cdot 47^{4} + \left(10 a + 41\right)\cdot 47^{5} + \left(4 a + 22\right)\cdot 47^{6} + \left(17 a + 10\right)\cdot 47^{7} + \left(5 a + 24\right)\cdot 47^{8} + \left(20 a + 13\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 39 + 23\cdot 47 + 7\cdot 47^{2} + 36\cdot 47^{3} + 35\cdot 47^{4} + 41\cdot 47^{5} + 9\cdot 47^{6} + 12\cdot 47^{7} + 37\cdot 47^{8} + 41\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 42 a + 34 + \left(41 a + 8\right)\cdot 47 + \left(15 a + 42\right)\cdot 47^{2} + \left(45 a + 25\right)\cdot 47^{3} + \left(42 a + 14\right)\cdot 47^{4} + \left(3 a + 42\right)\cdot 47^{5} + \left(35 a + 32\right)\cdot 47^{6} + \left(43 a + 16\right)\cdot 47^{7} + \left(45 a + 31\right)\cdot 47^{8} + \left(33 a + 16\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 6 a + 20 + \left(19 a + 9\right)\cdot 47 + \left(42 a + 17\right)\cdot 47^{2} + \left(a + 46\right)\cdot 47^{3} + \left(30 a + 33\right)\cdot 47^{4} + \left(44 a + 27\right)\cdot 47^{5} + \left(3 a + 34\right)\cdot 47^{6} + \left(3 a + 38\right)\cdot 47^{7} + \left(5 a + 23\right)\cdot 47^{8} + 45 a\cdot 47^{9} +O(47^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $4$ | $2$ | $(2,7)$ | $2$ |
| $4$ | $2$ | $(1,8)(2,7)(3,6)$ | $-2$ |
| $6$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(1,4)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $-2$ |
| $24$ | $2$ | $(1,4)(2,7)(5,8)$ | $0$ |
| $32$ | $3$ | $(2,3,4)(5,7,6)$ | $1$ |
| $12$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $12$ | $4$ | $(1,5,8,4)$ | $2$ |
| $12$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $-2$ |
| $24$ | $4$ | $(1,3)(2,5,7,4)(6,8)$ | $0$ |
| $24$ | $4$ | $(1,5,8,4)(2,7)$ | $0$ |
| $48$ | $4$ | $(1,2,3,4)(5,8,7,6)$ | $0$ |
| $32$ | $6$ | $(1,3,2,8,6,7)$ | $1$ |
| $32$ | $6$ | $(1,8)(2,3,4)(5,7,6)$ | $-1$ |
| $32$ | $6$ | $(1,8)(2,6,5,7,3,4)$ | $-1$ |
| $48$ | $8$ | $(1,4,3,2,8,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.