Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(855\)\(\medspace = 3^{2} \cdot 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.347236875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.475.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 139\cdot 191 + 188\cdot 191^{2} + 94\cdot 191^{3} + 27\cdot 191^{4} + 156\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 49 + 116\cdot 191 + 21\cdot 191^{2} + 34\cdot 191^{3} + 99\cdot 191^{4} + 24\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 73 + 145\cdot 191 + 93\cdot 191^{2} + 161\cdot 191^{3} + 179\cdot 191^{4} + 115\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 + 85\cdot 191 + 157\cdot 191^{2} + 140\cdot 191^{3} + 21\cdot 191^{4} + 144\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 108 + 135\cdot 191 + 118\cdot 191^{2} + 172\cdot 191^{3} + 17\cdot 191^{4} + 182\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 122 + 54\cdot 191 + 80\cdot 191^{2} + 66\cdot 191^{3} + 150\cdot 191^{4} + 84\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 153 + 134\cdot 191 + 164\cdot 191^{2} + 7\cdot 191^{3} + 95\cdot 191^{4} + 44\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 178 + 143\cdot 191 + 129\cdot 191^{2} + 85\cdot 191^{3} + 172\cdot 191^{4} + 11\cdot 191^{5} +O(191^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $0$ | $0$ |
| $4$ | $2$ | $(2,7)(4,8)(5,6)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,3,2)(4,8,6,5)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,5,7,4,3,8,2,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,2,5,3,6,7,8)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |