Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(322264470016\)\(\medspace = 2^{9} \cdot 857^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.2.4315322668808.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 16T1294 |
Parity: | odd |
Projective image: | $S_4\wr C_2$ |
Projective field: | Galois closure of 8.2.4315322668808.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{3} + x + 40 \)
Roots:
$r_{ 1 }$ | $=$ | \( 21 a + 10 + \left(22 a^{2} + a + 39\right)\cdot 43 + \left(26 a^{2} + 12 a + 39\right)\cdot 43^{2} + \left(3 a^{2} + 9 a + 6\right)\cdot 43^{3} + \left(29 a^{2} + 13 a + 31\right)\cdot 43^{4} + \left(39 a^{2} + 9 a + 20\right)\cdot 43^{5} + \left(24 a^{2} + 22 a + 18\right)\cdot 43^{6} + \left(34 a^{2} + 42 a + 22\right)\cdot 43^{7} + \left(16 a^{2} + 30 a + 38\right)\cdot 43^{8} + \left(5 a^{2} + 4 a + 1\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 2 }$ | $=$ | \( 14 + 12\cdot 43 + 19\cdot 43^{2} + 29\cdot 43^{3} + 7\cdot 43^{4} + 17\cdot 43^{5} + 37\cdot 43^{6} + 43^{7} + 4\cdot 43^{8} + 5\cdot 43^{9} +O(43^{10})\) |
$r_{ 3 }$ | $=$ | \( 40 + 2\cdot 43 + 20\cdot 43^{2} + 35\cdot 43^{3} + 41\cdot 43^{4} + 17\cdot 43^{5} + 33\cdot 43^{6} + 43^{7} + 16\cdot 43^{8} + 9\cdot 43^{9} +O(43^{10})\) |
$r_{ 4 }$ | $=$ | \( 3 a^{2} + 3 a + 12 + \left(2 a^{2} + 24 a + 40\right)\cdot 43 + \left(14 a^{2} + 8 a + 2\right)\cdot 43^{2} + \left(38 a^{2} + 41 a + 30\right)\cdot 43^{3} + \left(30 a^{2} + 11 a + 3\right)\cdot 43^{4} + \left(19 a^{2} + 5 a + 36\right)\cdot 43^{5} + \left(12 a^{2} + 27 a + 38\right)\cdot 43^{6} + \left(22 a^{2} + 31 a + 42\right)\cdot 43^{7} + \left(9 a^{2} + 16 a + 4\right)\cdot 43^{8} + \left(32 a^{2} + 18 a + 34\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 5 }$ | $=$ | \( 40 a^{2} + 19 a + 8 + \left(18 a^{2} + 17 a + 37\right)\cdot 43 + \left(2 a^{2} + 22 a + 23\right)\cdot 43^{2} + \left(a^{2} + 35 a + 19\right)\cdot 43^{3} + \left(26 a^{2} + 17 a\right)\cdot 43^{4} + \left(26 a^{2} + 28 a + 12\right)\cdot 43^{5} + \left(5 a^{2} + 36 a + 34\right)\cdot 43^{6} + \left(29 a^{2} + 11 a + 18\right)\cdot 43^{7} + \left(16 a^{2} + 38 a + 38\right)\cdot 43^{8} + \left(5 a^{2} + 19 a + 1\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 6 }$ | $=$ | \( 37 a^{2} + 42 a + 26 + \left(15 a^{2} + 22 a + 9\right)\cdot 43 + \left(30 a^{2} + 21 a + 42\right)\cdot 43^{2} + \left(36 a^{2} + 27 a + 26\right)\cdot 43^{3} + \left(8 a^{2} + 10 a + 20\right)\cdot 43^{4} + \left(13 a^{2} + 32 a + 31\right)\cdot 43^{5} + \left(18 a^{2} + 18 a + 29\right)\cdot 43^{6} + \left(40 a^{2} + 34 a + 40\right)\cdot 43^{7} + \left(29 a^{2} + 22 a + 28\right)\cdot 43^{8} + \left(28 a^{2} + 36 a + 1\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 7 }$ | $=$ | \( 30 a^{2} + 26 a + 7 + \left(3 a^{2} + a + 30\right)\cdot 43 + \left(30 a^{2} + 17 a + 27\right)\cdot 43^{2} + \left(11 a^{2} + 11 a + 24\right)\cdot 43^{3} + \left(14 a^{2} + 36 a + 38\right)\cdot 43^{4} + \left(40 a^{2} + 4 a + 20\right)\cdot 43^{5} + \left(19 a^{2} + 8 a + 16\right)\cdot 43^{6} + \left(4 a^{2} + 42 a + 2\right)\cdot 43^{7} + \left(22 a^{2} + 38 a + 38\right)\cdot 43^{8} + \left(34 a^{2} + 21 a + 19\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 8 }$ | $=$ | \( 19 a^{2} + 18 a + 14 + \left(23 a^{2} + 18 a\right)\cdot 43 + \left(25 a^{2} + 4 a + 39\right)\cdot 43^{2} + \left(37 a^{2} + 4 a + 41\right)\cdot 43^{3} + \left(19 a^{2} + 39 a + 27\right)\cdot 43^{4} + \left(32 a^{2} + 5 a + 15\right)\cdot 43^{5} + \left(4 a^{2} + 16 a + 6\right)\cdot 43^{6} + \left(41 a^{2} + 9 a + 41\right)\cdot 43^{7} + \left(33 a^{2} + 24 a + 2\right)\cdot 43^{8} + \left(22 a^{2} + 27 a + 12\right)\cdot 43^{9} +O(43^{10})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(3,7)(6,8)$ | $-3$ |
$9$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $1$ |
$12$ | $2$ | $(3,6)$ | $3$ |
$24$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $3$ |
$36$ | $2$ | $(1,2)(3,6)$ | $1$ |
$36$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
$16$ | $3$ | $(6,8,7)$ | $0$ |
$64$ | $3$ | $(2,5,4)(6,8,7)$ | $0$ |
$12$ | $4$ | $(3,6,7,8)$ | $-3$ |
$36$ | $4$ | $(1,2,4,5)(3,6,7,8)$ | $1$ |
$36$ | $4$ | $(1,2,4,5)(3,7)(6,8)$ | $1$ |
$72$ | $4$ | $(1,3,4,7)(2,6,5,8)$ | $-1$ |
$72$ | $4$ | $(1,2,4,5)(3,6)$ | $-1$ |
$144$ | $4$ | $(1,3,2,6)(4,7)(5,8)$ | $1$ |
$48$ | $6$ | $(1,4)(2,5)(6,7,8)$ | $0$ |
$96$ | $6$ | $(2,5,4)(3,6)$ | $0$ |
$192$ | $6$ | $(1,3)(2,6,5,8,4,7)$ | $0$ |
$144$ | $8$ | $(1,3,2,6,4,7,5,8)$ | $-1$ |
$96$ | $12$ | $(1,2,4,5)(6,8,7)$ | $0$ |