Basic invariants
| Dimension: | $6$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(875918098432\)\(\medspace = 2^{15} \cdot 13^{3} \cdot 23^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.2095196091449344.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T161 |
| Parity: | even |
| Determinant: | 1.2392.2t1.a.a |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective stem field: | Galois closure of 8.4.2095196091449344.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} - 40x^{6} + 16x^{5} + 656x^{4} + 1528x^{3} + 680x^{2} - 1008x - 2196 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{3} + x + 28 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a^{2} + 24 a + 19 + \left(28 a^{2} + 12 a + 28\right)\cdot 31 + \left(6 a^{2} + 15 a + 30\right)\cdot 31^{2} + \left(16 a^{2} + 15 a\right)\cdot 31^{3} + \left(11 a^{2} + 30 a + 12\right)\cdot 31^{4} + \left(16 a^{2} + 21 a + 23\right)\cdot 31^{5} + \left(29 a^{2} + 6 a + 1\right)\cdot 31^{6} + \left(2 a^{2} + 16 a\right)\cdot 31^{7} + \left(2 a^{2} + 24 a + 16\right)\cdot 31^{8} + 22 a^{2} 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 9 + \left(3 a^{2} + 9 a + 19\right)\cdot 31 + \left(22 a^{2} + 3 a + 7\right)\cdot 31^{2} + \left(26 a^{2} + 26 a + 19\right)\cdot 31^{3} + \left(23 a^{2} + 22 a + 18\right)\cdot 31^{4} + \left(19 a^{2} + 23 a + 11\right)\cdot 31^{5} + \left(23 a^{2} + 4 a + 19\right)\cdot 31^{6} + \left(12 a^{2} + 14 a + 5\right)\cdot 31^{7} + \left(16 a^{2} + 27\right)\cdot 31^{8} + \left(7 a^{2} + 4 a + 29\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 10\cdot 31 + 21\cdot 31^{2} + 26\cdot 31^{3} + 22\cdot 31^{4} + 4\cdot 31^{5} + 20\cdot 31^{6} + 8\cdot 31^{7} + 13\cdot 31^{8} + 18\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a^{2} + 9 a + 28 + \left(12 a^{2} + 19 a + 17\right)\cdot 31 + \left(23 a^{2} + 12 a\right)\cdot 31^{2} + \left(8 a^{2} + 20 a + 27\right)\cdot 31^{3} + \left(27 a^{2} + 29 a + 1\right)\cdot 31^{4} + \left(24 a^{2} + 5 a + 29\right)\cdot 31^{5} + \left(8 a^{2} + 12 a + 18\right)\cdot 31^{6} + \left(14 a^{2} + 29 a + 7\right)\cdot 31^{7} + \left(16 a^{2} + 9 a + 15\right)\cdot 31^{8} + \left(19 a^{2} + 25 a + 19\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( a^{2} + 25 a + 20 + \left(3 a^{2} + 28 a + 29\right)\cdot 31 + \left(18 a^{2} + 12 a + 4\right)\cdot 31^{2} + \left(21 a^{2} + 26 a + 26\right)\cdot 31^{3} + \left(a^{2} + 2 a + 3\right)\cdot 31^{4} + \left(8 a^{2} + 24 a + 14\right)\cdot 31^{5} + \left(13 a^{2} + a + 12\right)\cdot 31^{6} + \left(9 a^{2} + 6 a + 3\right)\cdot 31^{7} + \left(23 a^{2} + 25 a + 11\right)\cdot 31^{8} + \left(19 a^{2} + 25 a + 17\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a^{2} + 16 a + 29 + \left(24 a^{2} + 23 a + 2\right)\cdot 31 + \left(21 a^{2} + 14 a + 28\right)\cdot 31^{2} + \left(13 a^{2} + 9 a + 20\right)\cdot 31^{3} + \left(5 a^{2} + 5 a + 16\right)\cdot 31^{4} + \left(3 a^{2} + 14 a\right)\cdot 31^{5} + \left(25 a^{2} + 24 a + 10\right)\cdot 31^{6} + \left(8 a^{2} + 10 a + 13\right)\cdot 31^{7} + \left(22 a^{2} + 5 a + 10\right)\cdot 31^{8} + \left(3 a^{2} + a + 27\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 19 a^{2} + 29 a + 27 + \left(21 a^{2} + 29 a + 13\right)\cdot 31 + \left(2 a + 16\right)\cdot 31^{2} + \left(6 a^{2} + 26 a + 4\right)\cdot 31^{3} + \left(23 a^{2} + a + 30\right)\cdot 31^{4} + \left(20 a^{2} + 3 a + 15\right)\cdot 31^{5} + \left(23 a^{2} + 12 a + 18\right)\cdot 31^{6} + \left(13 a^{2} + 16 a + 17\right)\cdot 31^{7} + \left(12 a^{2} + 27 a + 12\right)\cdot 31^{8} + \left(20 a^{2} + 4 a + 30\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 + 31 + 14\cdot 31^{2} + 29\cdot 31^{3} + 17\cdot 31^{4} + 24\cdot 31^{5} + 22\cdot 31^{6} + 5\cdot 31^{7} + 18\cdot 31^{8} + 11\cdot 31^{9} +O(31^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $2$ |
| $9$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $36$ | $2$ | $(1,8)(5,6)$ | $-2$ |
| $16$ | $3$ | $(2,6,3)$ | $3$ |
| $32$ | $3$ | $(1,4,7)(2,3,5)$ | $0$ |
| $32$ | $3$ | $(1,4,7)(3,5,6)$ | $0$ |
| $36$ | $4$ | $(1,2,7,5)(3,8,6,4)$ | $0$ |
| $36$ | $4$ | $(1,3,7,6)(2,4,5,8)$ | $0$ |
| $36$ | $4$ | $(1,8,4,7)(2,5,3,6)$ | $2$ |
| $72$ | $4$ | $(1,8,4,7)(5,6)$ | $0$ |
| $48$ | $6$ | $(1,7)(2,6,3)(4,8)$ | $-1$ |
| $96$ | $6$ | $(1,3,4,5,7,2)(6,8)$ | $0$ |
| $96$ | $6$ | $(1,5,4,6,7,3)(2,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.