Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.2097152000.6 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1280.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 4x^{4} + 4x^{2} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 45\cdot 109 + 81\cdot 109^{2} + 92\cdot 109^{3} + 27\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 10\cdot 109 + 12\cdot 109^{2} + 9\cdot 109^{3} + 70\cdot 109^{4} + 50\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 99\cdot 109 + 5\cdot 109^{2} + 99\cdot 109^{3} + 12\cdot 109^{4} + 102\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 + 46\cdot 109 + 7\cdot 109^{2} + 56\cdot 109^{3} + 84\cdot 109^{4} + 79\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 68 + 62\cdot 109 + 101\cdot 109^{2} + 52\cdot 109^{3} + 24\cdot 109^{4} + 29\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 83 + 9\cdot 109 + 103\cdot 109^{2} + 9\cdot 109^{3} + 96\cdot 109^{4} + 6\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 87 + 98\cdot 109 + 96\cdot 109^{2} + 99\cdot 109^{3} + 38\cdot 109^{4} + 58\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 91 + 63\cdot 109 + 27\cdot 109^{2} + 16\cdot 109^{3} + 108\cdot 109^{4} + 81\cdot 109^{5} +O(109^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.