Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(307385777664000\)\(\medspace = 2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 7^{7} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.21346234560.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T274 |
| Parity: | odd |
| Determinant: | 1.35.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.21346234560.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 7x^{5} - 7x^{4} + 42x^{3} - 14x^{2} - 28x + 28 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 58 + \left(56 a + 47\right)\cdot 59 + \left(57 a + 14\right)\cdot 59^{2} + \left(31 a + 38\right)\cdot 59^{3} + \left(21 a + 40\right)\cdot 59^{4} + \left(53 a + 49\right)\cdot 59^{5} + \left(27 a + 29\right)\cdot 59^{6} + \left(57 a + 35\right)\cdot 59^{7} + \left(42 a + 27\right)\cdot 59^{8} + \left(11 a + 1\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 25\cdot 59 + 54\cdot 59^{2} + 12\cdot 59^{3} + 5\cdot 59^{4} + 36\cdot 59^{5} + 47\cdot 59^{6} + 52\cdot 59^{7} + 7\cdot 59^{8} + 8\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 53\cdot 59 + 6\cdot 59^{2} + 3\cdot 59^{3} + 30\cdot 59^{4} + 9\cdot 59^{5} + 41\cdot 59^{6} + 31\cdot 59^{7} + 14\cdot 59^{8} + 19\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + 56 + \left(57 a + 43\right)\cdot 59 + \left(27 a + 39\right)\cdot 59^{2} + \left(52 a + 56\right)\cdot 59^{3} + \left(35 a + 57\right)\cdot 59^{4} + \left(28 a + 21\right)\cdot 59^{5} + \left(52 a + 46\right)\cdot 59^{6} + \left(15 a + 27\right)\cdot 59^{7} + \left(7 a + 9\right)\cdot 59^{8} + \left(40 a + 53\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 a + 19 + \left(a + 20\right)\cdot 59 + \left(31 a + 10\right)\cdot 59^{2} + \left(6 a + 22\right)\cdot 59^{3} + \left(23 a + 41\right)\cdot 59^{4} + \left(30 a + 14\right)\cdot 59^{5} + \left(6 a + 11\right)\cdot 59^{6} + \left(43 a + 50\right)\cdot 59^{7} + 51 a\cdot 59^{8} + \left(18 a + 27\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 50 + \left(57 a + 19\right)\cdot 59 + \left(39 a + 55\right)\cdot 59^{2} + \left(57 a + 6\right)\cdot 59^{3} + \left(7 a + 40\right)\cdot 59^{4} + \left(54 a + 17\right)\cdot 59^{5} + \left(54 a + 27\right)\cdot 59^{6} + \left(50 a + 47\right)\cdot 59^{7} + \left(20 a + 7\right)\cdot 59^{8} + \left(30 a + 44\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 a + 16 + \left(a + 52\right)\cdot 59 + \left(19 a + 37\right)\cdot 59^{2} + \left(a + 24\right)\cdot 59^{3} + \left(51 a + 49\right)\cdot 59^{4} + \left(4 a + 4\right)\cdot 59^{5} + \left(4 a + 28\right)\cdot 59^{6} + \left(8 a + 43\right)\cdot 59^{7} + \left(38 a + 36\right)\cdot 59^{8} + \left(28 a + 53\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 46 a + 12 + \left(2 a + 32\right)\cdot 59 + \left(a + 16\right)\cdot 59^{2} + \left(27 a + 12\right)\cdot 59^{3} + \left(37 a + 30\right)\cdot 59^{4} + \left(5 a + 22\right)\cdot 59^{5} + \left(31 a + 4\right)\cdot 59^{6} + \left(a + 6\right)\cdot 59^{7} + \left(16 a + 13\right)\cdot 59^{8} + \left(47 a + 29\right)\cdot 59^{9} +O(59^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,6)(3,7)$ | $-3$ |
| $9$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $1$ |
| $12$ | $2$ | $(1,4)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-3$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,4)(2,6)(3,7)$ | $-1$ |
| $16$ | $3$ | $(1,5,8)$ | $0$ |
| $64$ | $3$ | $(1,5,8)(3,6,7)$ | $0$ |
| $12$ | $4$ | $(2,3,6,7)$ | $-3$ |
| $36$ | $4$ | $(1,4,5,8)(2,3,6,7)$ | $1$ |
| $36$ | $4$ | $(1,4,5,8)(2,6)(3,7)$ | $1$ |
| $72$ | $4$ | $(1,2,5,6)(3,8,7,4)$ | $1$ |
| $72$ | $4$ | $(1,4)(2,3,6,7)$ | $-1$ |
| $144$ | $4$ | $(1,3,4,2)(5,6)(7,8)$ | $-1$ |
| $48$ | $6$ | $(1,8,5)(2,6)(3,7)$ | $0$ |
| $96$ | $6$ | $(1,4)(3,7,6)$ | $0$ |
| $192$ | $6$ | $(1,3,5,6,8,7)(2,4)$ | $0$ |
| $144$ | $8$ | $(1,2,4,3,5,6,8,7)$ | $1$ |
| $96$ | $12$ | $(1,5,8)(2,3,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.