Basic invariants
Dimension: | $4$ |
Group: | $C_2 \wr S_4$ |
Conductor: | \(6008\)\(\medspace = 2^{3} \cdot 751 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.4512008.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2 \wr S_4$ |
Parity: | odd |
Determinant: | 1.6008.2t1.a.a |
Projective image: | $C_2^3:S_4$ |
Projective stem field: | Galois closure of 8.4.2310148096.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} + x^{6} - x^{5} + 2x^{4} - x^{3} + x^{2} + 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 10 a + 11 + 14\cdot 29 + \left(24 a + 9\right)\cdot 29^{2} + \left(6 a + 9\right)\cdot 29^{3} + \left(27 a + 21\right)\cdot 29^{4} + \left(15 a + 11\right)\cdot 29^{5} + \left(16 a + 5\right)\cdot 29^{6} + \left(a + 19\right)\cdot 29^{7} + \left(20 a + 15\right)\cdot 29^{8} + \left(24 a + 9\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 2 }$ |
$=$ |
\( a + 19 + \left(2 a + 28\right)\cdot 29 + \left(a + 11\right)\cdot 29^{2} + \left(3 a + 16\right)\cdot 29^{3} + \left(17 a + 20\right)\cdot 29^{4} + \left(8 a + 10\right)\cdot 29^{5} + \left(13 a + 1\right)\cdot 29^{6} + \left(17 a + 21\right)\cdot 29^{7} + \left(20 a + 19\right)\cdot 29^{8} + \left(22 a + 25\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 3 + 10\cdot 29 + 15\cdot 29^{2} + 23\cdot 29^{3} + 6\cdot 29^{4} + 6\cdot 29^{5} + 22\cdot 29^{6} + 25\cdot 29^{7} + 10\cdot 29^{8} + 3\cdot 29^{9} +O(29^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 24 a + 21 + \left(22 a + 19\right)\cdot 29 + \left(24 a + 8\right)\cdot 29^{2} + \left(19 a + 11\right)\cdot 29^{3} + \left(19 a + 5\right)\cdot 29^{4} + \left(18 a + 13\right)\cdot 29^{5} + \left(12 a + 9\right)\cdot 29^{6} + \left(26 a + 1\right)\cdot 29^{7} + 22\cdot 29^{8} + \left(24 a + 3\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 28 a + 24 + \left(26 a + 8\right)\cdot 29 + \left(27 a + 15\right)\cdot 29^{2} + \left(25 a + 1\right)\cdot 29^{3} + \left(11 a + 16\right)\cdot 29^{4} + \left(20 a + 7\right)\cdot 29^{5} + \left(15 a + 1\right)\cdot 29^{6} + \left(11 a + 8\right)\cdot 29^{7} + \left(8 a + 18\right)\cdot 29^{8} + \left(6 a + 2\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 10 + 5\cdot 29 + 19\cdot 29^{2} + 6\cdot 29^{3} + 14\cdot 29^{4} + 2\cdot 29^{5} + 8\cdot 29^{6} + 25\cdot 29^{7} + 29^{8} + 9\cdot 29^{9} +O(29^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 5 a + 25 + \left(6 a + 22\right)\cdot 29 + \left(4 a + 22\right)\cdot 29^{2} + \left(9 a + 27\right)\cdot 29^{3} + \left(9 a + 25\right)\cdot 29^{4} + \left(10 a + 28\right)\cdot 29^{5} + \left(16 a + 24\right)\cdot 29^{6} + \left(2 a + 4\right)\cdot 29^{7} + 28 a\cdot 29^{8} + \left(4 a + 7\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 19 a + 3 + \left(28 a + 6\right)\cdot 29 + \left(4 a + 13\right)\cdot 29^{2} + \left(22 a + 19\right)\cdot 29^{3} + \left(a + 5\right)\cdot 29^{4} + \left(13 a + 6\right)\cdot 29^{5} + \left(12 a + 14\right)\cdot 29^{6} + \left(27 a + 10\right)\cdot 29^{7} + \left(8 a + 27\right)\cdot 29^{8} + \left(4 a + 25\right)\cdot 29^{9} +O(29^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $4$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
$4$ | $2$ | $(2,7)$ | $2$ |
$4$ | $2$ | $(2,7)(3,6)(4,5)$ | $-2$ |
$6$ | $2$ | $(1,8)(2,7)$ | $0$ |
$12$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
$12$ | $2$ | $(1,3)(6,8)$ | $2$ |
$12$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $-2$ |
$24$ | $2$ | $(1,3)(2,7)(6,8)$ | $0$ |
$32$ | $3$ | $(2,3,4)(5,7,6)$ | $1$ |
$12$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
$12$ | $4$ | $(1,6,8,3)$ | $2$ |
$12$ | $4$ | $(1,8)(2,5,7,4)(3,6)$ | $-2$ |
$24$ | $4$ | $(1,2,8,7)(3,4)(5,6)$ | $0$ |
$24$ | $4$ | $(1,6,8,3)(2,7)$ | $0$ |
$48$ | $4$ | $(1,4,2,3)(5,7,6,8)$ | $0$ |
$32$ | $6$ | $(2,6,5,7,3,4)$ | $1$ |
$32$ | $6$ | $(1,8)(2,3,4)(5,7,6)$ | $-1$ |
$32$ | $6$ | $(1,8)(2,6,5,7,3,4)$ | $-1$ |
$48$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.