Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(518705353233\)\(\medspace = 3^{7} \cdot 619^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.6.321078613651227.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T272 |
Parity: | even |
Determinant: | 1.1857.2t1.a.a |
Projective image: | $S_4\wr C_2$ |
Projective stem field: | Galois closure of 8.6.321078613651227.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} - 14x^{6} + 17x^{5} + 43x^{4} + 323x^{3} + 310x^{2} - 1135x - 413 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 25 a + 28 + \left(8 a + 27\right)\cdot 29 + 25\cdot 29^{2} + \left(27 a + 14\right)\cdot 29^{3} + \left(19 a + 22\right)\cdot 29^{4} + \left(21 a + 2\right)\cdot 29^{5} + \left(25 a + 14\right)\cdot 29^{6} + 7 a\cdot 29^{7} + \left(4 a + 15\right)\cdot 29^{8} + \left(11 a + 10\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 2 }$ | $=$ | \( 14 + 10\cdot 29 + 11\cdot 29^{2} + 4\cdot 29^{3} + 8\cdot 29^{4} + 15\cdot 29^{5} + 15\cdot 29^{6} + 25\cdot 29^{7} + 21\cdot 29^{8} + 8\cdot 29^{9} +O(29^{10})\) |
$r_{ 3 }$ | $=$ | \( 28 a + 7 a\cdot 29 + \left(22 a + 13\right)\cdot 29^{2} + 17 a\cdot 29^{3} + \left(28 a + 9\right)\cdot 29^{4} + \left(3 a + 15\right)\cdot 29^{5} + \left(27 a + 11\right)\cdot 29^{6} + \left(28 a + 6\right)\cdot 29^{7} + \left(19 a + 15\right)\cdot 29^{8} + \left(26 a + 8\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 4 }$ | $=$ | \( a + 24 + \left(21 a + 11\right)\cdot 29 + 6 a\cdot 29^{2} + \left(11 a + 9\right)\cdot 29^{3} + 18\cdot 29^{4} + \left(25 a + 6\right)\cdot 29^{5} + \left(a + 27\right)\cdot 29^{6} + 7\cdot 29^{7} + \left(9 a + 28\right)\cdot 29^{8} + \left(2 a + 5\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 5 }$ | $=$ | \( 26 a + 19 + \left(18 a + 2\right)\cdot 29 + \left(19 a + 15\right)\cdot 29^{2} + \left(18 a + 18\right)\cdot 29^{3} + \left(5 a + 4\right)\cdot 29^{4} + \left(7 a + 21\right)\cdot 29^{5} + \left(6 a + 2\right)\cdot 29^{6} + \left(24 a + 10\right)\cdot 29^{7} + \left(8 a + 17\right)\cdot 29^{8} + \left(a + 10\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 6 }$ | $=$ | \( 23 + 2\cdot 29 + 24\cdot 29^{2} + 2\cdot 29^{4} + 28\cdot 29^{5} + 12\cdot 29^{6} + 13\cdot 29^{7} + 10\cdot 29^{8} + 29^{9} +O(29^{10})\) |
$r_{ 7 }$ | $=$ | \( 4 a + 8 + \left(20 a + 18\right)\cdot 29 + \left(28 a + 18\right)\cdot 29^{2} + \left(a + 4\right)\cdot 29^{3} + \left(9 a + 8\right)\cdot 29^{4} + \left(7 a + 4\right)\cdot 29^{5} + \left(3 a + 5\right)\cdot 29^{6} + \left(21 a + 14\right)\cdot 29^{7} + \left(24 a + 28\right)\cdot 29^{8} + \left(17 a + 3\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 8 }$ | $=$ | \( 3 a + 4 + \left(10 a + 13\right)\cdot 29 + \left(9 a + 7\right)\cdot 29^{2} + \left(10 a + 5\right)\cdot 29^{3} + \left(23 a + 14\right)\cdot 29^{4} + \left(21 a + 22\right)\cdot 29^{5} + \left(22 a + 26\right)\cdot 29^{6} + \left(4 a + 8\right)\cdot 29^{7} + \left(20 a + 8\right)\cdot 29^{8} + \left(27 a + 8\right)\cdot 29^{9} +O(29^{10})\) |
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 value |
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(1,4)(3,7)$ | $-3$ |
$9$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $1$ |
$12$ | $2$ | $(1,3)$ | $-3$ |
$24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $3$ |
$36$ | $2$ | $(1,3)(2,5)$ | $1$ |
$36$ | $2$ | $(1,3)(2,6)(5,8)$ | $1$ |
$16$ | $3$ | $(3,4,7)$ | $0$ |
$64$ | $3$ | $(3,4,7)(5,6,8)$ | $0$ |
$12$ | $4$ | $(1,3,4,7)$ | $3$ |
$36$ | $4$ | $(1,3,4,7)(2,5,6,8)$ | $1$ |
$36$ | $4$ | $(1,4)(2,5,6,8)(3,7)$ | $-1$ |
$72$ | $4$ | $(1,6,4,2)(3,8,7,5)$ | $-1$ |
$72$ | $4$ | $(1,3)(2,5,6,8)$ | $-1$ |
$144$ | $4$ | $(1,5,3,2)(4,6)(7,8)$ | $-1$ |
$48$ | $6$ | $(2,6)(3,7,4)(5,8)$ | $0$ |
$96$ | $6$ | $(1,3)(2,6,8)$ | $0$ |
$192$ | $6$ | $(1,2)(3,6,4,8,7,5)$ | $0$ |
$144$ | $8$ | $(1,5,3,6,4,8,7,2)$ | $1$ |
$96$ | $12$ | $(2,5,6,8)(3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.