Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(289\!\cdots\!072\)\(\medspace = 2^{63} \cdot 3^{22} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.9172942848.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.9172942848.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 16x^{5} + 13x^{4} + 8x^{3} - 14x^{2} - 4x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{3} + 3x + 42 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 39\cdot 47 + 11\cdot 47^{2} + 21\cdot 47^{3} + 40\cdot 47^{4} + 28\cdot 47^{5} + 45\cdot 47^{6} + 17\cdot 47^{7} + 4\cdot 47^{8} + 21\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a^{2} + a + 1 + \left(37 a^{2} + 38 a + 3\right)\cdot 47 + \left(16 a^{2} + 23 a + 16\right)\cdot 47^{2} + \left(43 a^{2} + 20 a + 3\right)\cdot 47^{3} + \left(42 a^{2} + 26 a + 14\right)\cdot 47^{4} + \left(34 a^{2} + 45 a + 18\right)\cdot 47^{5} + \left(41 a^{2} + 30 a + 28\right)\cdot 47^{6} + \left(41 a^{2} + 24 a + 25\right)\cdot 47^{7} + \left(29 a^{2} + 2 a + 18\right)\cdot 47^{8} + \left(36 a^{2} + 44 a + 22\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 a^{2} + 33 a + 15 + \left(11 a^{2} + 7 a + 45\right)\cdot 47 + \left(34 a^{2} + 4 a + 3\right)\cdot 47^{2} + \left(6 a^{2} + 42 a + 24\right)\cdot 47^{3} + \left(25 a^{2} + 24 a + 25\right)\cdot 47^{4} + \left(29 a^{2} + 21 a + 7\right)\cdot 47^{5} + \left(16 a^{2} + 27 a + 25\right)\cdot 47^{6} + \left(40 a^{2} + 32 a + 22\right)\cdot 47^{7} + \left(15 a^{2} + 11 a + 37\right)\cdot 47^{8} + \left(38 a^{2} + 29 a + 25\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a^{2} + 43 a + 38 + \left(37 a^{2} + 12 a + 45\right)\cdot 47 + \left(21 a^{2} + 17 a + 23\right)\cdot 47^{2} + \left(7 a^{2} + 27 a + 23\right)\cdot 47^{3} + \left(28 a^{2} + 41 a + 11\right)\cdot 47^{4} + \left(38 a^{2} + 32 a + 36\right)\cdot 47^{5} + \left(6 a^{2} + 29 a + 29\right)\cdot 47^{6} + \left(42 a^{2} + a + 15\right)\cdot 47^{7} + \left(20 a^{2} + 46 a + 40\right)\cdot 47^{8} + \left(38 a^{2} + 12 a + 22\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{2} + 13 a + 38 + \left(45 a^{2} + a + 18\right)\cdot 47 + \left(42 a^{2} + 19 a + 21\right)\cdot 47^{2} + \left(43 a^{2} + 31 a + 4\right)\cdot 47^{3} + \left(25 a^{2} + 42 a + 27\right)\cdot 47^{4} + \left(29 a^{2} + 26 a + 7\right)\cdot 47^{5} + \left(35 a^{2} + 35 a + 16\right)\cdot 47^{6} + \left(11 a^{2} + 36 a + 12\right)\cdot 47^{7} + \left(a^{2} + 32 a + 8\right)\cdot 47^{8} + \left(19 a^{2} + 20 a + 34\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 a^{2} + 12 a + \left(26 a^{2} + 38 a + 25\right)\cdot 47 + \left(46 a^{2} + 12 a + 26\right)\cdot 47^{2} + \left(2 a^{2} + 35 a + 14\right)\cdot 47^{3} + \left(25 a^{2} + 19 a + 5\right)\cdot 47^{4} + \left(29 a^{2} + 12 a + 18\right)\cdot 47^{5} + \left(40 a^{2} + 12 a + 3\right)\cdot 47^{6} + \left(22 a^{2} + 40 a + 24\right)\cdot 47^{7} + \left(42 a^{2} + 41 a + 36\right)\cdot 47^{8} + \left(28 a^{2} + 30 a + 3\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 a^{2} + 39 a + 27 + \left(29 a^{2} + 42 a + 30\right)\cdot 47 + \left(25 a^{2} + 16 a + 31\right)\cdot 47^{2} + \left(36 a^{2} + 31 a + 34\right)\cdot 47^{3} + \left(40 a^{2} + 32 a + 36\right)\cdot 47^{4} + \left(25 a^{2} + a + 10\right)\cdot 47^{5} + \left(46 a^{2} + 5 a + 15\right)\cdot 47^{6} + \left(28 a^{2} + 5 a + 36\right)\cdot 47^{7} + \left(30 a^{2} + 6 a + 12\right)\cdot 47^{8} + \left(26 a^{2} + 3 a + 46\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 42 + 26\cdot 47 + 5\cdot 47^{2} + 15\cdot 47^{3} + 27\cdot 47^{4} + 13\cdot 47^{5} + 24\cdot 47^{6} + 33\cdot 47^{7} + 29\cdot 47^{8} + 11\cdot 47^{9} +O(47^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,5)(3,8)$ | $-6$ |
| $9$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $2$ |
| $12$ | $2$ | $(1,4)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,4)(2,5)(3,8)$ | $0$ |
| $16$ | $3$ | $(1,6,7)$ | $0$ |
| $64$ | $3$ | $(1,6,7)(3,5,8)$ | $0$ |
| $12$ | $4$ | $(2,3,5,8)$ | $0$ |
| $36$ | $4$ | $(1,4,6,7)(2,3,5,8)$ | $-2$ |
| $36$ | $4$ | $(1,4,6,7)(2,5)(3,8)$ | $0$ |
| $72$ | $4$ | $(1,2,6,5)(3,7,8,4)$ | $0$ |
| $72$ | $4$ | $(1,4)(2,3,5,8)$ | $2$ |
| $144$ | $4$ | $(1,3,4,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,7,6)(2,5)(3,8)$ | $0$ |
| $96$ | $6$ | $(1,4)(3,8,5)$ | $0$ |
| $192$ | $6$ | $(1,3,6,5,7,8)(2,4)$ | $0$ |
| $144$ | $8$ | $(1,2,4,3,6,5,7,8)$ | $0$ |
| $96$ | $12$ | $(1,6,7)(2,3,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.