Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1368408064.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.68.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.272.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + x^{4} - 4 \)
|
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 29\cdot 53 + 48\cdot 53^{2} + 2\cdot 53^{3} + 24\cdot 53^{4} + 3\cdot 53^{5} + 43\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 25\cdot 53 + 3\cdot 53^{2} + 34\cdot 53^{3} + 45\cdot 53^{4} + 7\cdot 53^{5} + 3\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 10\cdot 53 + 38\cdot 53^{2} + 22\cdot 53^{3} + 51\cdot 53^{4} + 5\cdot 53^{5} + 20\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 52\cdot 53 + 26\cdot 53^{2} + 38\cdot 53^{3} + 50\cdot 53^{4} + 45\cdot 53^{5} + 29\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 26\cdot 53^{2} + 14\cdot 53^{3} + 2\cdot 53^{4} + 7\cdot 53^{5} + 23\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 38 + 42\cdot 53 + 14\cdot 53^{2} + 30\cdot 53^{3} + 53^{4} + 47\cdot 53^{5} + 32\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 44 + 27\cdot 53 + 49\cdot 53^{2} + 18\cdot 53^{3} + 7\cdot 53^{4} + 45\cdot 53^{5} + 49\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 48 + 23\cdot 53 + 4\cdot 53^{2} + 50\cdot 53^{3} + 28\cdot 53^{4} + 49\cdot 53^{5} + 9\cdot 53^{6} +O(53^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.