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 8.0.2097152000.4 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 12x^{4} - 12x^{2} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 10\cdot 61 + 31\cdot 61^{2} + 16\cdot 61^{3} + 48\cdot 61^{4} + 57\cdot 61^{5} + 38\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 5\cdot 61 + 30\cdot 61^{2} + 30\cdot 61^{3} + 32\cdot 61^{4} + 46\cdot 61^{5} + 9\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 6\cdot 61 + 45\cdot 61^{2} + 54\cdot 61^{3} + 18\cdot 61^{4} + 7\cdot 61^{5} +O(61^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 21\cdot 61 + 37\cdot 61^{2} + 8\cdot 61^{4} + 48\cdot 61^{5} + 5\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 + 39\cdot 61 + 23\cdot 61^{2} + 60\cdot 61^{3} + 52\cdot 61^{4} + 12\cdot 61^{5} + 55\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 43 + 54\cdot 61 + 15\cdot 61^{2} + 6\cdot 61^{3} + 42\cdot 61^{4} + 53\cdot 61^{5} + 60\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 49 + 55\cdot 61 + 30\cdot 61^{2} + 30\cdot 61^{3} + 28\cdot 61^{4} + 14\cdot 61^{5} + 51\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 58 + 50\cdot 61 + 29\cdot 61^{2} + 44\cdot 61^{3} + 12\cdot 61^{4} + 3\cdot 61^{5} + 22\cdot 61^{6} +O(61^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(3,6)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.