Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1412\)\(\medspace = 2^{2} \cdot 353 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.11260666112.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.5648.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 76\cdot 181 + 81\cdot 181^{2} + 139\cdot 181^{3} + 23\cdot 181^{4} + 19\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 74 + 60\cdot 181 + 149\cdot 181^{2} + 157\cdot 181^{3} + 92\cdot 181^{4} + 126\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 86 + 10\cdot 181 + 136\cdot 181^{2} + 53\cdot 181^{3} + 49\cdot 181^{4} + 176\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 112 + 78\cdot 181 + 45\cdot 181^{2} + 156\cdot 181^{3} + 149\cdot 181^{4} + 7\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 141 + 128\cdot 181 + 9\cdot 181^{2} + 32\cdot 181^{3} + 176\cdot 181^{4} + 145\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 149 + 122\cdot 181 + 113\cdot 181^{2} + 170\cdot 181^{3} + 116\cdot 181^{4} + 5\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 156 + 14\cdot 181 + 57\cdot 181^{2} + 144\cdot 181^{3} + 44\cdot 181^{4} + 116\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 162 + 50\cdot 181 + 131\cdot 181^{2} + 50\cdot 181^{3} + 70\cdot 181^{4} + 126\cdot 181^{5} +O(181^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(2,7)(4,5)(6,8)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,3,7)(4,5,6,8)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,8,2,4,3,5,7,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,8,3,6,2,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |