Basic invariants
| Dimension: | $9$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(426107828125\)\(\medspace = 5^{6} \cdot 7^{3} \cdot 43^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.205213530025.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T165 |
| Parity: | even |
| Determinant: | 1.301.2t1.a.a |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective stem field: | Galois closure of 8.4.205213530025.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 2x^{6} - 2x^{5} - 16x^{4} - 2x^{3} - 16x^{2} + 17x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{3} + 3x + 51 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a^{2} + 16 a + 10 + \left(12 a^{2} + 27 a + 14\right)\cdot 53 + \left(33 a^{2} + 50 a + 7\right)\cdot 53^{2} + \left(6 a^{2} + 33 a + 25\right)\cdot 53^{3} + \left(40 a^{2} + 38 a + 25\right)\cdot 53^{4} + \left(22 a^{2} + 38 a + 23\right)\cdot 53^{5} + \left(22 a^{2} + 11 a + 25\right)\cdot 53^{6} + \left(48 a^{2} + 46 a + 46\right)\cdot 53^{7} + \left(41 a^{2} + 26 a + 5\right)\cdot 53^{8} + \left(35 a^{2} + 10 a + 8\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 a^{2} + 30 a + 10 + \left(39 a^{2} + 7 a + 1\right)\cdot 53 + \left(38 a^{2} + 37 a + 44\right)\cdot 53^{2} + \left(24 a^{2} + 34 a + 6\right)\cdot 53^{3} + \left(38 a^{2} + 15 a + 48\right)\cdot 53^{4} + \left(44 a^{2} + 43 a + 34\right)\cdot 53^{5} + \left(28 a^{2} + 39 a + 22\right)\cdot 53^{6} + \left(16 a^{2} + 40 a + 4\right)\cdot 53^{7} + \left(6 a^{2} + 2 a + 24\right)\cdot 53^{8} + \left(9 a^{2} + 21 a + 3\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a^{2} + 24 a + 52 + \left(20 a^{2} + 43 a + 30\right)\cdot 53 + \left(50 a^{2} + 29 a + 41\right)\cdot 53^{2} + \left(32 a^{2} + 24 a + 24\right)\cdot 53^{3} + \left(18 a^{2} + 46 a + 35\right)\cdot 53^{4} + \left(34 a^{2} + 46\right)\cdot 53^{5} + \left(4 a^{2} + 38 a + 42\right)\cdot 53^{6} + \left(42 a^{2} + 32 a + 33\right)\cdot 53^{7} + \left(40 a^{2} + 8 a + 3\right)\cdot 53^{8} + \left(5 a^{2} + 19 a + 1\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a^{2} + 13 a + 19 + \left(19 a^{2} + 35 a + 29\right)\cdot 53 + \left(22 a^{2} + 25 a + 38\right)\cdot 53^{2} + \left(13 a^{2} + 47 a + 38\right)\cdot 53^{3} + \left(47 a^{2} + 20 a + 39\right)\cdot 53^{4} + \left(48 a^{2} + 13 a + 22\right)\cdot 53^{5} + \left(25 a^{2} + 3 a + 32\right)\cdot 53^{6} + \left(15 a^{2} + 27 a + 33\right)\cdot 53^{7} + \left(23 a^{2} + 17 a + 21\right)\cdot 53^{8} + \left(11 a^{2} + 23 a + 12\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 a^{2} + 23 a + 24 + \left(20 a^{2} + 2 a + 16\right)\cdot 53 + \left(43 a^{2} + 5 a\right)\cdot 53^{2} + \left(17 a^{2} + 28 a + 46\right)\cdot 53^{3} + \left(6 a^{2} + 44 a + 36\right)\cdot 53^{4} + \left(a^{2} + 15 a\right)\cdot 53^{5} + \left(34 a^{2} + 16 a + 33\right)\cdot 53^{6} + \left(29 a^{2} + 3 a + 30\right)\cdot 53^{7} + \left(40 a^{2} + 10 a + 39\right)\cdot 53^{8} + \left(43 a^{2} + 4 a + 19\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 38 + 21\cdot 53 + 47\cdot 53^{2} + 21\cdot 53^{3} + 33\cdot 53^{4} + 4\cdot 53^{5} + 52\cdot 53^{6} + 32\cdot 53^{7} + 18\cdot 53^{8} + 44\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 41 a^{2} + 34 + \left(45 a^{2} + 43 a + 13\right)\cdot 53 + \left(23 a^{2} + 10 a + 14\right)\cdot 53^{2} + \left(10 a^{2} + 43 a + 31\right)\cdot 53^{3} + \left(8 a^{2} + 45 a + 40\right)\cdot 53^{4} + \left(7 a^{2} + 46 a + 12\right)\cdot 53^{5} + \left(43 a^{2} + 49 a + 51\right)\cdot 53^{6} + \left(6 a^{2} + 8 a + 37\right)\cdot 53^{7} + \left(6 a^{2} + 40 a + 23\right)\cdot 53^{8} + \left(27 a + 38\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 25 + 31\cdot 53 + 18\cdot 53^{2} + 17\cdot 53^{3} + 5\cdot 53^{4} + 13\cdot 53^{5} + 5\cdot 53^{6} + 45\cdot 53^{7} + 21\cdot 53^{8} + 31\cdot 53^{9} +O(53^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,4)(3,8)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $1$ |
| $12$ | $2$ | $(1,5)(2,8)(3,6)(4,7)$ | $3$ |
| $12$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(4,8)(5,6)$ | $1$ |
| $16$ | $3$ | $(2,7,5)$ | $0$ |
| $32$ | $3$ | $(1,8,4)(2,7,5)$ | $0$ |
| $32$ | $3$ | $(1,8,3)(2,7,5)$ | $0$ |
| $36$ | $4$ | $(1,7,4,5)(2,8,6,3)$ | $-1$ |
| $36$ | $4$ | $(1,3,4,8)(2,7,6,5)$ | $1$ |
| $36$ | $4$ | $(1,6,4,2)(3,7,8,5)$ | $-1$ |
| $72$ | $4$ | $(1,8,4,3)(6,7)$ | $-1$ |
| $48$ | $6$ | $(1,4)(2,7,5)(3,8)$ | $0$ |
| $96$ | $6$ | $(1,5,8,2,4,7)(3,6)$ | $0$ |
| $96$ | $6$ | $(1,2,8,7,3,5)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.