Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.7247757312.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.3072.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 59^{2} + 33\cdot 59^{3} + 15\cdot 59^{4} + 28\cdot 59^{5} + 31\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 27\cdot 59 + 35\cdot 59^{2} + 16\cdot 59^{3} + 3\cdot 59^{4} + 54\cdot 59^{5} + 15\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 47\cdot 59 + 55\cdot 59^{2} + 54\cdot 59^{3} + 53\cdot 59^{4} + 56\cdot 59^{5} + 40\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 41\cdot 59 + 46\cdot 59^{2} + 51\cdot 59^{3} + 23\cdot 59^{4} + 12\cdot 59^{5} + 32\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 + 17\cdot 59 + 12\cdot 59^{2} + 7\cdot 59^{3} + 35\cdot 59^{4} + 46\cdot 59^{5} + 26\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 + 11\cdot 59 + 3\cdot 59^{2} + 4\cdot 59^{3} + 5\cdot 59^{4} + 2\cdot 59^{5} + 18\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 36 + 31\cdot 59 + 23\cdot 59^{2} + 42\cdot 59^{3} + 55\cdot 59^{4} + 4\cdot 59^{5} + 43\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 44 + 58\cdot 59 + 57\cdot 59^{2} + 25\cdot 59^{3} + 43\cdot 59^{4} + 30\cdot 59^{5} + 27\cdot 59^{6} +O(59^{7})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(3,6)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |