Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(2952\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.205797003264.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of \(\Q(\sqrt{9 +12 \sqrt{-2}})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 54\cdot 59 + 52\cdot 59^{2} + 7\cdot 59^{4} + 17\cdot 59^{5} + 28\cdot 59^{6} + 51\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 6\cdot 59 + 22\cdot 59^{2} + 18\cdot 59^{3} + 24\cdot 59^{4} + 58\cdot 59^{6} + 6\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 43\cdot 59^{2} + 6\cdot 59^{3} + 43\cdot 59^{4} + 25\cdot 59^{5} + 44\cdot 59^{6} + 9\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 31\cdot 59 + 22\cdot 59^{2} + 10\cdot 59^{3} + 18\cdot 59^{4} + 28\cdot 59^{5} + 33\cdot 59^{6} + 57\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 42\cdot 59 + 50\cdot 59^{2} + 27\cdot 59^{3} + 35\cdot 59^{4} + 11\cdot 59^{5} + 4\cdot 59^{6} + 13\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 + 21\cdot 59 + 30\cdot 59^{2} + 23\cdot 59^{3} + 32\cdot 59^{4} + 4\cdot 59^{5} + 41\cdot 59^{6} + 43\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 47 + 47\cdot 59 + 29\cdot 59^{2} + 13\cdot 59^{3} + 15\cdot 59^{4} + 15\cdot 59^{5} + 53\cdot 59^{6} + 30\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 49 + 32\cdot 59 + 43\cdot 59^{2} + 16\cdot 59^{3} + 59^{4} + 15\cdot 59^{5} + 32\cdot 59^{6} + 22\cdot 59^{7} +O(59^{8})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,6)(3,4)(5,8)$ | $0$ | $0$ |
| $4$ | $2$ | $(2,4)(3,8)(6,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,5,4)(3,8,6,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,6,4,8,5,3,2,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,2,6,5,7,4,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |