Basic invariants
| Dimension: | $8$ |
| Group: | $C_2 \wr S_4$ |
| Conductor: | \(1635798918612635401\)\(\medspace = 29^{6} \cdot 229^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.1520789.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 24T708 |
| Parity: | even |
| Projective image: | $C_2^3:S_4$ |
| Projective field: | Galois closure of 8.4.37090522921.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 22 + \left(7 a + 33\right)\cdot 47 + \left(32 a + 11\right)\cdot 47^{2} + \left(7 a + 12\right)\cdot 47^{3} + \left(15 a + 16\right)\cdot 47^{4} + \left(36 a + 28\right)\cdot 47^{5} + \left(28 a + 22\right)\cdot 47^{6} + \left(39 a + 43\right)\cdot 47^{7} + \left(42 a + 10\right)\cdot 47^{8} + \left(10 a + 36\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 26\cdot 47 + 28\cdot 47^{2} + 26\cdot 47^{3} + 28\cdot 47^{4} + 17\cdot 47^{5} + 47^{6} + 18\cdot 47^{7} + 41\cdot 47^{8} + 8\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 a + 24 + \left(44 a + 43\right)\cdot 47 + \left(34 a + 13\right)\cdot 47^{2} + \left(21 a + 11\right)\cdot 47^{3} + \left(14 a + 32\right)\cdot 47^{4} + \left(35 a + 24\right)\cdot 47^{5} + \left(15 a + 30\right)\cdot 47^{6} + \left(21 a + 1\right)\cdot 47^{7} + \left(32 a + 40\right)\cdot 47^{8} + \left(2 a + 2\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 a + 21 + \left(6 a + 26\right)\cdot 47 + \left(34 a + 30\right)\cdot 47^{2} + \left(33 a + 42\right)\cdot 47^{3} + \left(43 a + 36\right)\cdot 47^{4} + \left(42 a + 19\right)\cdot 47^{5} + \left(8 a + 19\right)\cdot 47^{6} + \left(25 a + 46\right)\cdot 47^{7} + \left(32 a + 45\right)\cdot 47^{8} + \left(41 a + 22\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 44 + \left(40 a + 4\right)\cdot 47 + \left(12 a + 45\right)\cdot 47^{2} + \left(13 a + 28\right)\cdot 47^{3} + \left(3 a + 43\right)\cdot 47^{4} + \left(4 a + 14\right)\cdot 47^{5} + \left(38 a + 41\right)\cdot 47^{6} + \left(21 a + 40\right)\cdot 47^{7} + \left(14 a + 38\right)\cdot 47^{8} + \left(5 a + 26\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 a + 28 + \left(39 a + 44\right)\cdot 47 + \left(14 a + 21\right)\cdot 47^{2} + \left(39 a + 42\right)\cdot 47^{3} + \left(31 a + 38\right)\cdot 47^{4} + \left(10 a + 38\right)\cdot 47^{5} + \left(18 a + 43\right)\cdot 47^{6} + \left(7 a + 46\right)\cdot 47^{7} + \left(4 a + 9\right)\cdot 47^{8} + \left(36 a + 15\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 17 + 15\cdot 47 + 44\cdot 47^{2} + 3\cdot 47^{3} + 46\cdot 47^{4} + 9\cdot 47^{5} + 2\cdot 47^{6} + 9\cdot 47^{7} + 11\cdot 47^{8} + 5\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( a + 22 + \left(2 a + 40\right)\cdot 47 + \left(12 a + 38\right)\cdot 47^{2} + \left(25 a + 19\right)\cdot 47^{3} + \left(32 a + 39\right)\cdot 47^{4} + \left(11 a + 33\right)\cdot 47^{5} + \left(31 a + 26\right)\cdot 47^{6} + \left(25 a + 28\right)\cdot 47^{7} + \left(14 a + 36\right)\cdot 47^{8} + \left(44 a + 22\right)\cdot 47^{9} +O(47^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-8$ |
| $4$ | $2$ | $(2,7)$ | $-4$ |
| $4$ | $2$ | $(1,8)(2,7)(3,6)$ | $4$ |
| $6$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(1,4)(5,8)$ | $0$ |
| $12$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $24$ | $2$ | $(1,4)(2,7)(5,8)$ | $0$ |
| $32$ | $3$ | $(1,3,2)(6,7,8)$ | $-1$ |
| $12$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $12$ | $4$ | $(1,4,8,5)$ | $0$ |
| $12$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $0$ |
| $24$ | $4$ | $(1,3)(2,5,7,4)(6,8)$ | $0$ |
| $24$ | $4$ | $(1,4,8,5)(2,7)$ | $0$ |
| $48$ | $4$ | $(1,4,3,2)(5,6,7,8)$ | $0$ |
| $32$ | $6$ | $(1,3,2,8,6,7)$ | $1$ |
| $32$ | $6$ | $(1,3,2)(4,5)(6,7,8)$ | $-1$ |
| $32$ | $6$ | $(1,3,2,8,6,7)(4,5)$ | $1$ |
| $48$ | $8$ | $(1,4,3,2,8,5,6,7)$ | $0$ |