Basic invariants
| Dimension: | $6$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(603709440\)\(\medspace = 2^{13} \cdot 3 \cdot 5 \cdot 17^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.82104483840.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.2040.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.82104483840.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} - 4x^{5} - 18x^{4} - 8x^{3} - 40x^{2} + 44x - 15 \)
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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 $\mathbb{Q}_{ 47 }$:
\( x^{3} + 3x + 42 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 37 a^{2} + 14 a + 37 + \left(45 a^{2} + 25 a + 7\right)\cdot 47 + \left(11 a^{2} + 24 a + 3\right)\cdot 47^{2} + \left(12 a^{2} + 29 a + 40\right)\cdot 47^{3} + \left(40 a^{2} + 5 a + 41\right)\cdot 47^{4} + \left(19 a^{2} + 34 a + 22\right)\cdot 47^{5} + \left(2 a^{2} + 39 a + 18\right)\cdot 47^{6} + \left(38 a^{2} + 15 a + 2\right)\cdot 47^{7} + \left(42 a^{2} + 45 a + 45\right)\cdot 47^{8} + \left(17 a^{2} + 17 a + 11\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a^{2} + 33 a + 26 + \left(42 a^{2} + 25 a + 12\right)\cdot 47 + \left(36 a^{2} + 19 a + 22\right)\cdot 47^{2} + \left(4 a^{2} + 33 a + 12\right)\cdot 47^{3} + \left(10 a^{2} + 33 a + 33\right)\cdot 47^{4} + \left(29 a^{2} + 20 a + 21\right)\cdot 47^{5} + \left(11 a^{2} + 35 a + 43\right)\cdot 47^{6} + \left(45 a^{2} + 8 a + 3\right)\cdot 47^{7} + \left(a^{2} + 43 a + 24\right)\cdot 47^{8} + \left(40 a^{2} + 22 a + 39\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 16\cdot 47 + 15\cdot 47^{2} + 22\cdot 47^{4} + 3\cdot 47^{5} + 6\cdot 47^{6} + 33\cdot 47^{7} + 27\cdot 47^{8} + 24\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a^{2} + 19 a + 40 + \left(33 a^{2} + 30 a + 41\right)\cdot 47 + \left(16 a^{2} + 21 a + 28\right)\cdot 47^{2} + \left(3 a^{2} + 22 a + 9\right)\cdot 47^{3} + \left(5 a^{2} + 33 a + 23\right)\cdot 47^{4} + \left(37 a^{2} + 23 a + 37\right)\cdot 47^{5} + \left(6 a^{2} + 10 a + 33\right)\cdot 47^{6} + \left(17 a^{2} + 37 a + 41\right)\cdot 47^{7} + \left(2 a^{2} + 28 a + 24\right)\cdot 47^{8} + \left(6 a + 6\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 + 28\cdot 47 + 13\cdot 47^{2} + 38\cdot 47^{3} + 7\cdot 47^{4} + 16\cdot 47^{5} + 33\cdot 47^{6} + 24\cdot 47^{7} + 33\cdot 47^{8} + 27\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 a^{2} + 14 a + 31 + \left(39 a^{2} + 8 a + 42\right)\cdot 47 + \left(a^{2} + 13 a + 29\right)\cdot 47^{2} + \left(42 a^{2} + 12 a + 5\right)\cdot 47^{3} + \left(43 a^{2} + 46 a + 2\right)\cdot 47^{4} + \left(21 a^{2} + a + 27\right)\cdot 47^{5} + \left(32 a^{2} + 32 a + 31\right)\cdot 47^{6} + \left(18 a^{2} + 7 a + 10\right)\cdot 47^{7} + \left(14 a^{2} + 7 a + 35\right)\cdot 47^{8} + \left(11 a^{2} + 5 a + 45\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 a^{2} + 19 a + 9 + \left(8 a^{2} + 13 a + 27\right)\cdot 47 + \left(33 a^{2} + 9 a + 45\right)\cdot 47^{2} + \left(39 a^{2} + 5 a\right)\cdot 47^{3} + \left(9 a^{2} + 42 a + 28\right)\cdot 47^{4} + \left(5 a^{2} + 10 a + 40\right)\cdot 47^{5} + \left(12 a^{2} + 22 a + 37\right)\cdot 47^{6} + \left(37 a^{2} + 23 a\right)\cdot 47^{7} + \left(36 a^{2} + 41 a + 33\right)\cdot 47^{8} + \left(17 a^{2} + 23 a + 11\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 a^{2} + 42 a + 22 + \left(18 a^{2} + 37 a + 11\right)\cdot 47 + \left(40 a^{2} + 5 a + 29\right)\cdot 47^{2} + \left(38 a^{2} + 38 a + 33\right)\cdot 47^{3} + \left(31 a^{2} + 26 a + 29\right)\cdot 47^{4} + \left(27 a^{2} + 2 a + 18\right)\cdot 47^{5} + \left(28 a^{2} + a + 30\right)\cdot 47^{6} + \left(31 a^{2} + a + 23\right)\cdot 47^{7} + \left(42 a^{2} + 22 a + 11\right)\cdot 47^{8} + \left(6 a^{2} + 17 a + 20\right)\cdot 47^{9} +O(47^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $6$ | $2$ | $(2,5)(4,8)$ | $2$ |
| $9$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,3)$ | $4$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $2$ |
| $36$ | $2$ | $(1,3)(2,5)(4,8)$ | $0$ |
| $16$ | $3$ | $(1,6,7)$ | $3$ |
| $64$ | $3$ | $(1,6,7)(4,5,8)$ | $0$ |
| $12$ | $4$ | $(2,4,5,8)$ | $2$ |
| $36$ | $4$ | $(1,3,6,7)(2,4,5,8)$ | $-2$ |
| $36$ | $4$ | $(1,3,6,7)(2,5)(4,8)$ | $-2$ |
| $72$ | $4$ | $(1,2,6,5)(3,4,7,8)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,4,5,8)$ | $0$ |
| $144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,7,6)(2,5)(4,8)$ | $-1$ |
| $96$ | $6$ | $(1,3)(4,8,5)$ | $1$ |
| $192$ | $6$ | $(1,4,6,5,7,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,4,6,5,7,8)$ | $0$ |
| $96$ | $12$ | $(1,6,7)(2,4,5,8)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.