Basic invariants
| Dimension: | $9$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(830584000000\)\(\medspace = 2^{9} \cdot 5^{6} \cdot 47^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.499679334400.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T165 |
| Parity: | even |
| Determinant: | 1.376.2t1.a.a |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective stem field: | Galois closure of 8.4.499679334400.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 18x^{6} - 14x^{5} + 65x^{4} + 126x^{3} + 99x^{2} + 112x + 64 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{3} + 5x + 57 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 a^{2} + 9 a + 35 + \left(14 a^{2} + 50 a + 55\right)\cdot 59 + \left(16 a^{2} + 53 a + 35\right)\cdot 59^{2} + \left(2 a^{2} + 13 a + 33\right)\cdot 59^{3} + \left(14 a^{2} + 21 a + 9\right)\cdot 59^{4} + \left(13 a^{2} + 53 a + 54\right)\cdot 59^{5} + \left(5 a^{2} + 7 a + 6\right)\cdot 59^{6} + \left(14 a^{2} + 54 a + 42\right)\cdot 59^{7} + \left(50 a^{2} + 19 a + 16\right)\cdot 59^{8} + \left(27 a^{2} + a + 37\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 a^{2} + 12 a + 33 + \left(11 a^{2} + 3 a + 23\right)\cdot 59 + \left(a^{2} + 46 a + 22\right)\cdot 59^{2} + \left(17 a^{2} + 56 a + 36\right)\cdot 59^{3} + \left(55 a + 31\right)\cdot 59^{4} + \left(7 a^{2} + 26 a + 5\right)\cdot 59^{5} + \left(a^{2} + 56 a + 56\right)\cdot 59^{6} + \left(27 a^{2} + 45 a + 56\right)\cdot 59^{7} + \left(37 a^{2} + 44 a + 24\right)\cdot 59^{8} + \left(38 a^{2} + 12 a + 19\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 a^{2} + 56 a + 33 + \left(8 a^{2} + 32 a + 13\right)\cdot 59 + \left(7 a^{2} + 14 a + 42\right)\cdot 59^{2} + \left(23 a^{2} + 34 a + 56\right)\cdot 59^{3} + \left(43 a^{2} + 7 a + 17\right)\cdot 59^{4} + \left(11 a^{2} + 22 a + 21\right)\cdot 59^{5} + \left(37 a^{2} + 44 a + 58\right)\cdot 59^{6} + \left(37 a^{2} + 56 a + 52\right)\cdot 59^{7} + \left(24 a^{2} + 19 a + 1\right)\cdot 59^{8} + \left(6 a^{2} + 50 a + 30\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a^{2} + 50 a + 17 + \left(38 a^{2} + 22 a + 35\right)\cdot 59 + \left(50 a^{2} + 57 a + 49\right)\cdot 59^{2} + \left(18 a^{2} + 26 a + 22\right)\cdot 59^{3} + \left(15 a^{2} + 54 a + 42\right)\cdot 59^{4} + \left(40 a^{2} + 9 a + 37\right)\cdot 59^{5} + \left(20 a^{2} + 17 a + 42\right)\cdot 59^{6} + \left(53 a^{2} + 15 a + 26\right)\cdot 59^{7} + \left(55 a^{2} + 53 a + 27\right)\cdot 59^{8} + \left(13 a^{2} + 54 a + 35\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 + 45\cdot 59 + 3\cdot 59^{2} + 2\cdot 59^{3} + 26\cdot 59^{4} + 53\cdot 59^{5} + 19\cdot 59^{6} + 40\cdot 59^{7} + 4\cdot 59^{8} + 33\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 a^{2} + 57 a + 46 + \left(21 a^{2} + 29 a + 40\right)\cdot 59 + \left(3 a^{2} + 49 a + 12\right)\cdot 59^{2} + \left(13 a^{2} + 56 a + 30\right)\cdot 59^{3} + \left(23 a^{2} + 40 a + 20\right)\cdot 59^{4} + \left(7 a^{2} + 28 a + 54\right)\cdot 59^{5} + \left(11 a^{2} + 42 a + 6\right)\cdot 59^{6} + \left(56 a^{2} + 34 a + 25\right)\cdot 59^{7} + \left(37 a^{2} + 18 a + 54\right)\cdot 59^{8} + \left(18 a^{2} + 50 a + 45\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 + 35\cdot 59 + 54\cdot 59^{2} + 39\cdot 59^{3} + 52\cdot 59^{4} + 28\cdot 59^{5} + 31\cdot 59^{6} + 14\cdot 59^{7} + 39\cdot 59^{8} + 48\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 52 a + 11 + \left(23 a^{2} + 37 a + 45\right)\cdot 59 + \left(39 a^{2} + 14 a + 14\right)\cdot 59^{2} + \left(43 a^{2} + 47 a + 14\right)\cdot 59^{3} + \left(21 a^{2} + 55 a + 35\right)\cdot 59^{4} + \left(38 a^{2} + 35 a + 39\right)\cdot 59^{5} + \left(42 a^{2} + 8 a + 13\right)\cdot 59^{6} + \left(47 a^{2} + 29 a + 36\right)\cdot 59^{7} + \left(29 a^{2} + 20 a + 7\right)\cdot 59^{8} + \left(12 a^{2} + 7 a + 45\right)\cdot 59^{9} +O(59^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,7)(6,8)$ | $-3$ |
| $9$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $1$ |
| $12$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $3$ |
| $12$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $3$ |
| $36$ | $2$ | $(4,5)(6,7)$ | $1$ |
| $16$ | $3$ | $(2,4,5)$ | $0$ |
| $32$ | $3$ | $(1,7,8)(2,4,5)$ | $0$ |
| $32$ | $3$ | $(1,8,6)(2,5,4)$ | $0$ |
| $36$ | $4$ | $(1,6,7,8)(2,5,4,3)$ | $1$ |
| $36$ | $4$ | $(1,4,7,2)(3,6,5,8)$ | $-1$ |
| $36$ | $4$ | $(1,3,7,5)(2,6,4,8)$ | $-1$ |
| $72$ | $4$ | $(2,5,4,3)(7,8)$ | $-1$ |
| $48$ | $6$ | $(1,7)(2,4,5)(6,8)$ | $0$ |
| $96$ | $6$ | $(1,2,7,4,8,5)(3,6)$ | $0$ |
| $96$ | $6$ | $(1,4,8,2,6,5)(3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.