Basic invariants
| Dimension: | $9$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(746636341248\)\(\medspace = 2^{12} \cdot 3^{12} \cdot 7^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.3186376704.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T165 |
| Parity: | even |
| Determinant: | 1.28.2t1.a.a |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective stem field: | Galois closure of 8.4.3186376704.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 4x^{6} + 4x^{5} + 4x^{4} + 4x^{3} + 4x^{2} - 8x - 6 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a^{2} + 19 a + 22 + \left(5 a^{2} + 7 a + 2\right)\cdot 29 + \left(15 a^{2} + 19 a + 16\right)\cdot 29^{2} + \left(14 a^{2} + 23 a + 1\right)\cdot 29^{3} + \left(9 a + 15\right)\cdot 29^{4} + \left(22 a^{2} + a + 24\right)\cdot 29^{5} + \left(14 a^{2} + 16 a + 7\right)\cdot 29^{6} + \left(24 a^{2} + 13 a + 11\right)\cdot 29^{7} + \left(10 a^{2} + 21 a + 22\right)\cdot 29^{8} + \left(23 a^{2} + 15 a + 17\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 27\cdot 29 + 2\cdot 29^{2} + 27\cdot 29^{3} + 22\cdot 29^{4} + 24\cdot 29^{5} + 22\cdot 29^{6} + 10\cdot 29^{7} + 27\cdot 29^{8} + 5\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a^{2} + 5 + \left(28 a^{2} + 17 a + 16\right)\cdot 29 + \left(2 a^{2} + 28 a + 23\right)\cdot 29^{2} + \left(8 a^{2} + 28 a + 11\right)\cdot 29^{3} + \left(15 a^{2} + 24 a + 8\right)\cdot 29^{4} + \left(13 a^{2} + 17 a + 11\right)\cdot 29^{5} + \left(6 a + 2\right)\cdot 29^{6} + \left(26 a^{2} + 12 a + 10\right)\cdot 29^{7} + \left(27 a^{2} + 10 a + 22\right)\cdot 29^{8} + \left(9 a^{2} + 5 a + 20\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 a^{2} + 9 a + 3 + \left(a^{2} + 16 a + 26\right)\cdot 29 + \left(15 a + 24\right)\cdot 29^{2} + \left(10 a^{2} + 22 a + 14\right)\cdot 29^{3} + \left(20 a^{2} + 10 a + 12\right)\cdot 29^{4} + \left(17 a^{2} + 15 a + 28\right)\cdot 29^{5} + \left(8 a^{2} + 2 a + 18\right)\cdot 29^{6} + \left(6 a^{2} + 26 a + 25\right)\cdot 29^{7} + \left(9 a^{2} + 2 a\right)\cdot 29^{8} + \left(14 a^{2} + 27 a + 25\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 20 + \left(6 a^{2} + 11 a + 14\right)\cdot 29 + \left(26 a^{2} + 25\right)\cdot 29^{2} + \left(4 a^{2} + 28 a + 26\right)\cdot 29^{3} + \left(28 a^{2} + 19 a + 15\right)\cdot 29^{4} + \left(15 a^{2} + 13 a + 14\right)\cdot 29^{5} + \left(8 a^{2} + 2 a + 3\right)\cdot 29^{6} + \left(3 a^{2} + 27 a + 28\right)\cdot 29^{7} + \left(5 a^{2} + 15 a + 20\right)\cdot 29^{8} + \left(19 a^{2} + 5 a + 13\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a^{2} + 12 a + 6 + \left(23 a^{2} + 18\right)\cdot 29 + \left(28 a^{2} + 9\right)\cdot 29^{2} + \left(15 a^{2} + a + 22\right)\cdot 29^{3} + \left(14 a^{2} + 13 a + 26\right)\cdot 29^{4} + \left(28 a^{2} + 26 a + 11\right)\cdot 29^{5} + \left(19 a^{2} + 19 a + 28\right)\cdot 29^{6} + \left(28 a^{2} + 18 a + 3\right)\cdot 29^{7} + \left(24 a^{2} + 2 a + 28\right)\cdot 29^{8} + \left(28 a^{2} + 18 a + 16\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 a^{2} + a + 24 + \left(21 a^{2} + 5 a + 13\right)\cdot 29 + \left(13 a^{2} + 23 a + 4\right)\cdot 29^{2} + \left(4 a^{2} + 11 a + 17\right)\cdot 29^{3} + \left(8 a^{2} + 8 a + 15\right)\cdot 29^{4} + \left(18 a^{2} + 12 a + 19\right)\cdot 29^{5} + \left(5 a^{2} + 10 a + 24\right)\cdot 29^{6} + \left(27 a^{2} + 18 a + 14\right)\cdot 29^{7} + \left(8 a^{2} + 4 a\right)\cdot 29^{8} + \left(20 a^{2} + 15 a + 4\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 22 + 25\cdot 29 + 8\cdot 29^{2} + 23\cdot 29^{3} + 27\cdot 29^{4} + 9\cdot 29^{5} + 7\cdot 29^{6} + 11\cdot 29^{7} + 22\cdot 29^{8} + 11\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(3,6)(5,8)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $1$ |
| $12$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $3$ |
| $12$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $3$ |
| $36$ | $2$ | $(1,7)(5,8)$ | $1$ |
| $16$ | $3$ | $(3,8,6)$ | $0$ |
| $32$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $32$ | $3$ | $(1,2,4)(5,6,8)$ | $0$ |
| $36$ | $4$ | $(1,3,4,6)(2,5,7,8)$ | $-1$ |
| $36$ | $4$ | $(1,5,4,8)(2,6,7,3)$ | $-1$ |
| $36$ | $4$ | $(1,7,4,2)(3,5,6,8)$ | $1$ |
| $72$ | $4$ | $(3,5,6,8)(4,7)$ | $-1$ |
| $48$ | $6$ | $(1,4)(2,7)(3,8,6)$ | $0$ |
| $96$ | $6$ | $(1,5,2,6,4,3)(7,8)$ | $0$ |
| $96$ | $6$ | $(1,6,2,8,4,5)(3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.