Basic invariants
| Dimension: | $9$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(681472000000\)\(\medspace = 2^{15} \cdot 5^{6} \cdot 11^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.4.599695360000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T165 |
| Parity: | even |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective field: | Galois closure of 8.4.599695360000.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{3} + 5x + 57 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a^{2} + 13 a + 8 + \left(30 a^{2} + 48 a + 17\right)\cdot 59 + \left(50 a^{2} + 47 a + 52\right)\cdot 59^{2} + \left(49 a^{2} + 6 a + 19\right)\cdot 59^{3} + \left(24 a^{2} + 26 a + 47\right)\cdot 59^{4} + \left(27 a^{2} + a + 1\right)\cdot 59^{5} + \left(37 a^{2} + 20 a + 44\right)\cdot 59^{6} + \left(42 a^{2} + 38 a + 53\right)\cdot 59^{7} + \left(16 a^{2} + 26 a + 19\right)\cdot 59^{8} + \left(21 a^{2} + 34 a + 29\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a^{2} + 39 a + 34 + \left(54 a^{2} + 22 a + 12\right)\cdot 59 + \left(22 a^{2} + 2 a + 14\right)\cdot 59^{2} + \left(10 a^{2} + 31 a + 56\right)\cdot 59^{3} + \left(54 a^{2} + 19 a + 37\right)\cdot 59^{4} + \left(13 a^{2} + 14 a + 17\right)\cdot 59^{5} + \left(55 a^{2} + 2 a + 47\right)\cdot 59^{6} + \left(9 a^{2} + 52 a + 49\right)\cdot 59^{7} + \left(45 a^{2} + 16 a + 51\right)\cdot 59^{8} + \left(30 a^{2} + 58 a + 10\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 44 a^{2} + 23 a + 9 + \left(44 a^{2} + 30 a + 45\right)\cdot 59 + \left(39 a^{2} + 49 a + 55\right)\cdot 59^{2} + \left(22 a^{2} + 7\right)\cdot 59^{3} + \left(5 a^{2} + 51 a + 2\right)\cdot 59^{4} + \left(19 a^{2} + 34 a + 33\right)\cdot 59^{5} + \left(10 a^{2} + 32 a + 12\right)\cdot 59^{6} + \left(5 a^{2} + 10 a + 27\right)\cdot 59^{7} + \left(3 a^{2} + 26 a + 33\right)\cdot 59^{8} + \left(46 a^{2} + 44 a + 13\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 39\cdot 59 + 23\cdot 59^{2} + 28\cdot 59^{3} + 5\cdot 59^{4} + 54\cdot 59^{5} + 7\cdot 59^{6} + 36\cdot 59^{7} + 32\cdot 59^{8} + 22\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a^{2} + 29 a + 28 + \left(5 a^{2} + 48 a + 46\right)\cdot 59 + \left(7 a^{2} + 57 a\right)\cdot 59^{2} + \left(17 a^{2} + 36 a\right)\cdot 59^{3} + \left(29 a^{2} + 48 a + 14\right)\cdot 59^{4} + \left(28 a^{2} + 40 a + 7\right)\cdot 59^{5} + \left(24 a^{2} + 48 a + 43\right)\cdot 59^{6} + \left(55 a^{2} + 5 a + 4\right)\cdot 59^{7} + \left(54 a^{2} + 56 a + 45\right)\cdot 59^{8} + \left(35 a^{2} + 55 a + 47\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 48 a^{2} + 23 a + 42 + \left(42 a^{2} + 39 a + 38\right)\cdot 59 + \left(27 a^{2} + 20 a + 15\right)\cdot 59^{2} + \left(45 a^{2} + 51 a + 5\right)\cdot 59^{3} + \left(28 a^{2} + 40 a + 21\right)\cdot 59^{4} + \left(12 a^{2} + 22 a + 50\right)\cdot 59^{5} + \left(11 a^{2} + 6 a + 54\right)\cdot 59^{6} + \left(11 a^{2} + 10 a + 7\right)\cdot 59^{7} + \left(39 a^{2} + 6 a + 16\right)\cdot 59^{8} + \left(50 a^{2} + 39 a + 9\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 55 + 7\cdot 59 + 39\cdot 59^{2} + 50\cdot 59^{3} + 56\cdot 59^{4} + 5\cdot 59^{5} + 55\cdot 59^{6} + 59^{7} + 17\cdot 59^{8} + 23\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 43 a^{2} + 50 a + 10 + \left(58 a^{2} + 46 a + 28\right)\cdot 59 + \left(28 a^{2} + 57 a + 34\right)\cdot 59^{2} + \left(31 a^{2} + 49 a + 8\right)\cdot 59^{3} + \left(34 a^{2} + 49 a + 51\right)\cdot 59^{4} + \left(16 a^{2} + 3 a + 6\right)\cdot 59^{5} + \left(38 a^{2} + 8 a + 30\right)\cdot 59^{6} + \left(52 a^{2} + a + 54\right)\cdot 59^{7} + \left(17 a^{2} + 45 a + 19\right)\cdot 59^{8} + \left(51 a^{2} + 3 a + 20\right)\cdot 59^{9} +O(59^{10})\)
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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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,4)(3,6)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $1$ |
| $12$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $3$ |
| $12$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $3$ |
| $36$ | $2$ | $(4,6)(7,8)$ | $1$ |
| $16$ | $3$ | $(1,6,3)$ | $0$ |
| $32$ | $3$ | $(1,4,6)(2,7,8)$ | $0$ |
| $32$ | $3$ | $(1,4,6)(5,7,8)$ | $0$ |
| $36$ | $4$ | $(1,7,4,2)(3,8,6,5)$ | $-1$ |
| $36$ | $4$ | $(1,5,4,8)(2,3,7,6)$ | $-1$ |
| $36$ | $4$ | $(1,4,3,6)(2,8,5,7)$ | $1$ |
| $72$ | $4$ | $(2,8,5,7)(4,6)$ | $-1$ |
| $48$ | $6$ | $(1,6,3)(2,7)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,7,4,8,6,2)(3,5)$ | $0$ |
| $96$ | $6$ | $(1,5,4,7,6,8)(2,3)$ | $0$ |