Basic invariants
| Dimension: | $9$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(592704000000\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.4.1244678400.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T165 |
| Parity: | even |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective field: | Galois closure of 8.4.1244678400.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{3} + 6x + 35 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a^{2} + 3 a + 27 + \left(a + 26\right)\cdot 37 + \left(14 a^{2} + a + 35\right)\cdot 37^{2} + \left(31 a^{2} + 22 a + 15\right)\cdot 37^{3} + \left(3 a^{2} + 11 a + 2\right)\cdot 37^{4} + \left(5 a^{2} + 24 a + 24\right)\cdot 37^{5} + \left(20 a^{2} + 18 a + 32\right)\cdot 37^{6} + \left(12 a^{2} + 5 a + 4\right)\cdot 37^{7} + \left(29 a^{2} + 15 a + 26\right)\cdot 37^{8} + \left(7 a^{2} + 16 a + 1\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 28\cdot 37 + 16\cdot 37^{2} + 13\cdot 37^{3} + 17\cdot 37^{4} + 34\cdot 37^{5} + 37^{6} + 23\cdot 37^{7} + 21\cdot 37^{8} + 28\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 36\cdot 37 + 23\cdot 37^{2} + 32\cdot 37^{3} + 37^{4} + 26\cdot 37^{5} + 32\cdot 37^{6} + 24\cdot 37^{7} + 14\cdot 37^{8} + 14\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 31 a + 2 + \left(15 a^{2} + 31 a + 14\right)\cdot 37 + \left(10 a^{2} + 29 a + 11\right)\cdot 37^{2} + \left(36 a^{2} + 23 a + 17\right)\cdot 37^{3} + \left(13 a^{2} + 24 a + 25\right)\cdot 37^{4} + \left(5 a^{2} + 3 a + 34\right)\cdot 37^{5} + \left(2 a^{2} + 9 a + 7\right)\cdot 37^{6} + \left(6 a^{2} + 4 a + 4\right)\cdot 37^{7} + \left(36 a^{2} + 5 a + 14\right)\cdot 37^{8} + \left(30 a^{2} + 36 a + 3\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a^{2} + 2 a + 10 + \left(18 a^{2} + 14 a + 25\right)\cdot 37 + \left(30 a^{2} + 7 a + 27\right)\cdot 37^{2} + \left(23 a^{2} + 22 a + 22\right)\cdot 37^{3} + \left(20 a^{2} + 18 a + 32\right)\cdot 37^{4} + \left(36 a^{2} + 30 a + 1\right)\cdot 37^{5} + \left(13 a^{2} + 18 a + 8\right)\cdot 37^{6} + \left(9 a^{2} + 11 a + 29\right)\cdot 37^{7} + \left(5 a^{2} + 36 a + 3\right)\cdot 37^{8} + \left(8 a^{2} + 27 a + 3\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a^{2} + 32 a + 13 + \left(17 a^{2} + 21 a + 22\right)\cdot 37 + \left(29 a^{2} + 28 a + 23\right)\cdot 37^{2} + \left(18 a^{2} + 29 a + 2\right)\cdot 37^{3} + \left(12 a^{2} + 6 a\right)\cdot 37^{4} + \left(32 a^{2} + 19 a + 22\right)\cdot 37^{5} + \left(2 a^{2} + 36 a\right)\cdot 37^{6} + \left(15 a^{2} + 19 a + 15\right)\cdot 37^{7} + \left(2 a^{2} + 22 a + 29\right)\cdot 37^{8} + \left(21 a^{2} + 29 a + 17\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 a^{2} + 14 a + 33 + \left(15 a^{2} + a + 15\right)\cdot 37 + \left(29 a^{2} + 10 a + 13\right)\cdot 37^{2} + \left(6 a^{2} + 27 a + 10\right)\cdot 37^{3} + \left(6 a^{2} + 5 a + 31\right)\cdot 37^{4} + \left(33 a^{2} + 6 a + 34\right)\cdot 37^{5} + \left(25 a^{2} + 23 a + 28\right)\cdot 37^{6} + \left(19 a^{2} + 24 a + 21\right)\cdot 37^{7} + \left(35 a^{2} + 6 a + 11\right)\cdot 37^{8} + \left(5 a^{2} + 35 a + 14\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 8 a^{2} + 29 a + 10 + \left(6 a^{2} + 3 a + 15\right)\cdot 37 + \left(34 a^{2} + 34 a + 32\right)\cdot 37^{2} + \left(30 a^{2} + 22 a + 32\right)\cdot 37^{3} + \left(16 a^{2} + 6 a + 36\right)\cdot 37^{4} + \left(35 a^{2} + 27 a + 6\right)\cdot 37^{5} + \left(8 a^{2} + 4 a + 35\right)\cdot 37^{6} + \left(11 a^{2} + 8 a + 24\right)\cdot 37^{7} + \left(2 a^{2} + 25 a + 26\right)\cdot 37^{8} + \left(2 a + 27\right)\cdot 37^{9} +O(37^{10})\)
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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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,7)(4,8)$ | $-3$ |
| $9$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $1$ |
| $12$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $3$ |
| $12$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $3$ |
| $36$ | $2$ | $(3,5)(7,8)$ | $1$ |
| $16$ | $3$ | $(1,6,3)$ | $0$ |
| $32$ | $3$ | $(1,6,3)(2,8,7)$ | $0$ |
| $32$ | $3$ | $(1,6,3)(2,8,4)$ | $0$ |
| $36$ | $4$ | $(1,8,5,4)(2,6,7,3)$ | $-1$ |
| $36$ | $4$ | $(1,6,5,3)(2,4,7,8)$ | $1$ |
| $36$ | $4$ | $(1,2,5,7)(3,4,6,8)$ | $-1$ |
| $72$ | $4$ | $(2,8,7,4)(5,6)$ | $-1$ |
| $48$ | $6$ | $(1,6,3)(2,7)(4,8)$ | $0$ |
| $96$ | $6$ | $(1,7,6,2,3,8)(4,5)$ | $0$ |
| $96$ | $6$ | $(1,8,6,4,3,2)(5,7)$ | $0$ |