Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.2097152000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.40.4t1.b.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.2000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{4} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 11\cdot 109 + 77\cdot 109^{3} + 19\cdot 109^{4} + 18\cdot 109^{5} + 84\cdot 109^{6} +O(109^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 108\cdot 109 + 92\cdot 109^{2} + 2\cdot 109^{3} + 41\cdot 109^{4} + 21\cdot 109^{5} + 48\cdot 109^{6} +O(109^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 109 + 24\cdot 109^{2} + 77\cdot 109^{3} + 58\cdot 109^{4} + 6\cdot 109^{5} + 75\cdot 109^{6} +O(109^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 + 42\cdot 109 + 74\cdot 109^{2} + 39\cdot 109^{3} + 89\cdot 109^{4} + 74\cdot 109^{5} + 53\cdot 109^{6} +O(109^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 62 + 66\cdot 109 + 34\cdot 109^{2} + 69\cdot 109^{3} + 19\cdot 109^{4} + 34\cdot 109^{5} + 55\cdot 109^{6} +O(109^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 73 + 107\cdot 109 + 84\cdot 109^{2} + 31\cdot 109^{3} + 50\cdot 109^{4} + 102\cdot 109^{5} + 33\cdot 109^{6} +O(109^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 84 + 16\cdot 109^{2} + 106\cdot 109^{3} + 67\cdot 109^{4} + 87\cdot 109^{5} + 60\cdot 109^{6} +O(109^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 98 + 97\cdot 109 + 108\cdot 109^{2} + 31\cdot 109^{3} + 89\cdot 109^{4} + 90\cdot 109^{5} + 24\cdot 109^{6} +O(109^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,4,7,5)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,5,7,4)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,3,8,6)(2,7)(4,5)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6,8,3)(2,7)(4,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $4$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $0$ |
| $4$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.