Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(885027826179\)\(\medspace = 3^{4} \cdot 7^{3} \cdot 317^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.5725588181607.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.2219.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.5725588181607.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 24x^{6} - 14x^{5} + 48x^{4} - 59x^{3} - 13x^{2} + 18x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 17\cdot 61 + 6\cdot 61^{2} + 24\cdot 61^{3} + 50\cdot 61^{4} + 25\cdot 61^{5} + 13\cdot 61^{6} + 27\cdot 61^{7} + 22\cdot 61^{8} + 49\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 a + 22 + \left(21 a + 58\right)\cdot 61 + \left(11 a + 5\right)\cdot 61^{2} + \left(47 a + 26\right)\cdot 61^{3} + 15 a\cdot 61^{4} + \left(60 a + 47\right)\cdot 61^{5} + \left(46 a + 8\right)\cdot 61^{6} + \left(50 a + 30\right)\cdot 61^{7} + \left(48 a + 7\right)\cdot 61^{8} + \left(15 a + 20\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 a + 58 + \left(28 a + 1\right)\cdot 61 + 36 a\cdot 61^{2} + \left(46 a + 42\right)\cdot 61^{3} + \left(22 a + 29\right)\cdot 61^{4} + \left(35 a + 3\right)\cdot 61^{5} + \left(60 a + 41\right)\cdot 61^{6} + \left(45 a + 47\right)\cdot 61^{7} + \left(a + 30\right)\cdot 61^{8} + \left(58 a + 19\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 32 + \left(32 a + 56\right)\cdot 61 + \left(24 a + 7\right)\cdot 61^{2} + \left(14 a + 52\right)\cdot 61^{3} + \left(38 a + 5\right)\cdot 61^{4} + \left(25 a + 16\right)\cdot 61^{5} + 5\cdot 61^{6} + \left(15 a + 33\right)\cdot 61^{7} + \left(59 a + 47\right)\cdot 61^{8} + \left(2 a + 14\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 45 + \left(23 a + 35\right)\cdot 61 + \left(30 a + 56\right)\cdot 61^{2} + \left(43 a + 40\right)\cdot 61^{3} + \left(38 a + 17\right)\cdot 61^{4} + 21 a\cdot 61^{5} + 39\cdot 61^{6} + \left(53 a + 2\right)\cdot 61^{7} + \left(2 a + 49\right)\cdot 61^{8} + \left(57 a + 60\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 + 46\cdot 61 + 46\cdot 61^{2} + 3\cdot 61^{3} + 36\cdot 61^{4} + 15\cdot 61^{5} + 61^{6} + 14\cdot 61^{7} + 21\cdot 61^{8} + 38\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a + 55 + \left(39 a + 46\right)\cdot 61 + \left(49 a + 56\right)\cdot 61^{2} + 13 a\cdot 61^{3} + \left(45 a + 30\right)\cdot 61^{4} + 30\cdot 61^{5} + \left(14 a + 56\right)\cdot 61^{6} + \left(10 a + 33\right)\cdot 61^{7} + \left(12 a + 5\right)\cdot 61^{8} + \left(45 a + 48\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 44 a + 1 + \left(37 a + 42\right)\cdot 61 + \left(30 a + 2\right)\cdot 61^{2} + \left(17 a + 54\right)\cdot 61^{3} + \left(22 a + 12\right)\cdot 61^{4} + \left(39 a + 44\right)\cdot 61^{5} + \left(60 a + 17\right)\cdot 61^{6} + \left(7 a + 55\right)\cdot 61^{7} + \left(58 a + 59\right)\cdot 61^{8} + \left(3 a + 53\right)\cdot 61^{9} +O(61^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,7)(5,8)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $1$ |
| $12$ | $2$ | $(1,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $3$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $1$ |
| $36$ | $2$ | $(1,3)(2,7)(5,8)$ | $-1$ |
| $16$ | $3$ | $(1,4,6)$ | $0$ |
| $64$ | $3$ | $(1,4,6)(5,7,8)$ | $0$ |
| $12$ | $4$ | $(2,5,7,8)$ | $-3$ |
| $36$ | $4$ | $(1,3,4,6)(2,5,7,8)$ | $1$ |
| $36$ | $4$ | $(1,3,4,6)(2,7)(5,8)$ | $1$ |
| $72$ | $4$ | $(1,2,4,7)(3,5,6,8)$ | $-1$ |
| $72$ | $4$ | $(1,3)(2,5,7,8)$ | $-1$ |
| $144$ | $4$ | $(1,5,3,2)(4,7)(6,8)$ | $1$ |
| $48$ | $6$ | $(1,6,4)(2,7)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(5,8,7)$ | $0$ |
| $192$ | $6$ | $(1,5,4,7,6,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,5,4,7,6,8)$ | $-1$ |
| $96$ | $12$ | $(1,4,6)(2,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.