Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1767\)\(\medspace = 3 \cdot 19 \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.16551253989.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.5301.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 50\cdot 181 + 180\cdot 181^{2} + 78\cdot 181^{3} + 95\cdot 181^{4} + 106\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 + 101\cdot 181 + 163\cdot 181^{2} + 9\cdot 181^{3} + 117\cdot 181^{4} + 83\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 139\cdot 181 + 34\cdot 181^{2} + 22\cdot 181^{3} + 180\cdot 181^{4} + 131\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 + 70\cdot 181 + 4\cdot 181^{2} + 88\cdot 181^{3} + 167\cdot 181^{4} + 79\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 48 + 38\cdot 181 + 158\cdot 181^{2} + 111\cdot 181^{3} + 144\cdot 181^{4} + 70\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 88 + 125\cdot 181 + 62\cdot 181^{2} + 142\cdot 181^{3} + 118\cdot 181^{4} + 74\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 113 + 49\cdot 181 + 75\cdot 181^{2} + 57\cdot 181^{3} + 7\cdot 181^{4} + 56\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 159 + 149\cdot 181 + 44\cdot 181^{2} + 32\cdot 181^{3} + 74\cdot 181^{4} + 120\cdot 181^{5} +O(181^{6})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,8)(3,7)(5,6)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,5,8)(2,3,4,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,2,8,7,5,4,6,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,7,6,2,5,3,8,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |