Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Artin stem field: | Galois closure of 8.4.5151653888.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.136.4t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.39304.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{5} - 14x^{4} + 16x^{3} + 32x^{2} + 16x + 2 \)
|
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 40\cdot 127 + 107\cdot 127^{2} + 19\cdot 127^{3} + 75\cdot 127^{4} + 84\cdot 127^{5} + 20\cdot 127^{6} + 36\cdot 127^{7} + 124\cdot 127^{8} + 31\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 36\cdot 127 + 42\cdot 127^{2} + 59\cdot 127^{3} + 47\cdot 127^{4} + 30\cdot 127^{5} + 77\cdot 127^{6} + 13\cdot 127^{7} + 18\cdot 127^{8} + 40\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 32 + 4\cdot 127 + 46\cdot 127^{2} + 127^{3} + 3\cdot 127^{4} + 2\cdot 127^{5} + 57\cdot 127^{6} + 79\cdot 127^{7} + 69\cdot 127^{8} + 59\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 + 121\cdot 127 + 44\cdot 127^{2} + 63\cdot 127^{3} + 101\cdot 127^{4} + 81\cdot 127^{5} + 105\cdot 127^{6} + 66\cdot 127^{7} + 28\cdot 127^{8} + 18\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 79 + 3\cdot 127 + 75\cdot 127^{2} + 3\cdot 127^{3} + 72\cdot 127^{4} + 41\cdot 127^{5} + 76\cdot 127^{6} + 85\cdot 127^{7} + 104\cdot 127^{8} + 19\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 80 + 124\cdot 127 + 87\cdot 127^{2} + 58\cdot 127^{3} + 77\cdot 127^{4} + 127^{5} + 15\cdot 127^{6} + 22\cdot 127^{7} + 51\cdot 127^{8} + 29\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 103 + 94\cdot 127 + 13\cdot 127^{2} + 112\cdot 127^{3} + 126\cdot 127^{4} + 85\cdot 127^{5} + 112\cdot 127^{6} + 127^{7} + 50\cdot 127^{8} + 47\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 117 + 82\cdot 127 + 90\cdot 127^{2} + 62\cdot 127^{3} + 4\cdot 127^{4} + 53\cdot 127^{5} + 43\cdot 127^{6} + 75\cdot 127^{7} + 61\cdot 127^{8} + 7\cdot 127^{9} +O(127^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(2,8)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,3)(4,7)(5,8)$ | $0$ |
| $1$ | $4$ | $(1,5,7,3)(2,6,8,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,7,5)(2,4,8,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,7,5)(2,6,8,4)$ | $0$ |
| $2$ | $4$ | $(2,6,8,4)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,4,8,6)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,3,7,5)(2,8)(4,6)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,5,7,3)(2,8)(4,6)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,4,7,6)(2,3,8,5)$ | $0$ |
| $4$ | $8$ | $(1,8,5,4,7,2,3,6)$ | $0$ |
| $4$ | $8$ | $(1,4,3,8,7,6,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.