Basic invariants
| Dimension: | $9$ |
| Group: | $(A_4\wr C_2):C_2$ |
| Conductor: | \(836962177024\)\(\medspace = 2^{12} \cdot 19^{3} \cdot 31^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.30810670141696.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T165 |
| Parity: | even |
| Determinant: | 1.589.2t1.a.a |
| Projective image: | $\PGOPlus(4,3)$ |
| Projective stem field: | Galois closure of 8.4.30810670141696.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} - 4x^{6} + 8x^{5} + 51x^{4} - 6x^{3} - 60x^{2} - 76x - 57 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{3} + 3x + 42 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 45 a^{2} + 13 a + 9 + \left(28 a^{2} + 6 a + 12\right)\cdot 47 + \left(12 a^{2} + 40 a + 21\right)\cdot 47^{2} + \left(46 a^{2} + 33 a + 30\right)\cdot 47^{3} + \left(5 a^{2} + 2 a + 7\right)\cdot 47^{4} + \left(46 a^{2} + 35 a + 14\right)\cdot 47^{5} + \left(23 a^{2} + 35 a + 2\right)\cdot 47^{6} + \left(32 a^{2} + 30 a + 26\right)\cdot 47^{7} + \left(38 a^{2} + 42 a + 33\right)\cdot 47^{8} + \left(22 a^{2} + 26 a + 22\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 43 a^{2} + 3 a + 16 + \left(20 a^{2} + 13 a + 26\right)\cdot 47 + \left(35 a^{2} + 21 a + 11\right)\cdot 47^{2} + \left(26 a^{2} + 39 a + 8\right)\cdot 47^{3} + \left(24 a^{2} + 42 a + 45\right)\cdot 47^{4} + \left(25 a^{2} + 28 a + 7\right)\cdot 47^{5} + \left(12 a^{2} + 17 a + 22\right)\cdot 47^{6} + \left(19 a^{2} + 18 a + 28\right)\cdot 47^{7} + \left(24 a^{2} + 37 a + 38\right)\cdot 47^{8} + \left(6 a^{2} + 19 a + 27\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 a^{2} + 24 a + 16 + \left(5 a^{2} + 15 a + 43\right)\cdot 47 + \left(20 a^{2} + 34 a + 27\right)\cdot 47^{2} + \left(33 a^{2} + 31 a + 21\right)\cdot 47^{3} + \left(11 a^{2} + 39 a + 19\right)\cdot 47^{4} + \left(21 a^{2} + 15 a + 46\right)\cdot 47^{5} + \left(22 a^{2} + 14 a + 41\right)\cdot 47^{6} + \left(38 a^{2} + a + 19\right)\cdot 47^{7} + \left(17 a^{2} + 6 a + 25\right)\cdot 47^{8} + \left(21 a^{2} + 45 a + 10\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 43\cdot 47 + 11\cdot 47^{2} + 45\cdot 47^{3} + 12\cdot 47^{4} + 46\cdot 47^{5} + 42\cdot 47^{6} + 22\cdot 47^{7} + 37\cdot 47^{8} + 21\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 a^{2} + 13 a + 28 + \left(25 a^{2} + 7 a + 5\right)\cdot 47 + \left(7 a^{2} + 33 a + 11\right)\cdot 47^{2} + \left(37 a^{2} + 23 a + 12\right)\cdot 47^{3} + \left(29 a^{2} + 25 a + 8\right)\cdot 47^{4} + \left(23 a^{2} + 28 a + 16\right)\cdot 47^{5} + \left(10 a^{2} + 19 a + 22\right)\cdot 47^{6} + \left(42 a^{2} + 20 a + 45\right)\cdot 47^{7} + \left(44 a^{2} + 34 a + 45\right)\cdot 47^{8} + \left(39 a^{2} + a + 9\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 + 46\cdot 47 + 36\cdot 47^{2} + 41\cdot 47^{3} + 11\cdot 47^{4} + 35\cdot 47^{5} + 8\cdot 47^{6} + 30\cdot 47^{7} + 30\cdot 47^{8} + 2\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 18 a^{2} + 21 a + 2 + \left(39 a^{2} + 33 a + 33\right)\cdot 47 + \left(26 a^{2} + 20 a + 2\right)\cdot 47^{2} + \left(10 a^{2} + 36 a + 6\right)\cdot 47^{3} + \left(11 a^{2} + 18 a + 18\right)\cdot 47^{4} + \left(24 a^{2} + 30 a + 17\right)\cdot 47^{5} + \left(12 a^{2} + 38 a + 26\right)\cdot 47^{6} + \left(19 a^{2} + 42 a + 46\right)\cdot 47^{7} + \left(10 a^{2} + 16 a + 23\right)\cdot 47^{8} + \left(31 a^{2} + 18 a + 39\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 8 a^{2} + 20 a + 40 + \left(20 a^{2} + 18 a + 24\right)\cdot 47 + \left(38 a^{2} + 38 a + 17\right)\cdot 47^{2} + \left(33 a^{2} + 22 a + 22\right)\cdot 47^{3} + \left(10 a^{2} + 11 a + 17\right)\cdot 47^{4} + \left(2 a + 4\right)\cdot 47^{5} + \left(12 a^{2} + 15 a + 21\right)\cdot 47^{6} + \left(36 a^{2} + 27 a + 15\right)\cdot 47^{7} + \left(4 a^{2} + 3 a + 46\right)\cdot 47^{8} + \left(19 a^{2} + 29 a + 5\right)\cdot 47^{9} +O(47^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,6)(3,8)$ | $-3$ |
| $9$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $1$ |
| $12$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $3$ |
| $12$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(2,8)(4,5)$ | $1$ |
| $16$ | $3$ | $(1,7,5)$ | $0$ |
| $32$ | $3$ | $(1,5,7)(3,6,8)$ | $0$ |
| $32$ | $3$ | $(1,7,5)(2,8,6)$ | $0$ |
| $36$ | $4$ | $(1,8,5,3)(2,7,6,4)$ | $-1$ |
| $36$ | $4$ | $(1,7,5,4)(2,3,6,8)$ | $1$ |
| $36$ | $4$ | $(1,6,5,2)(3,4,8,7)$ | $-1$ |
| $72$ | $4$ | $(2,8,6,3)(4,5)$ | $-1$ |
| $48$ | $6$ | $(1,7,5)(2,6)(3,8)$ | $0$ |
| $96$ | $6$ | $(1,8,5,3,7,6)(2,4)$ | $0$ |
| $96$ | $6$ | $(1,8,7,6,5,2)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.