Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(784\)\(\medspace = 2^{4} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.843308032.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.1372.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 74\cdot 197 + 187\cdot 197^{2} + 73\cdot 197^{3} + 4\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 104\cdot 197 + 182\cdot 197^{2} + 107\cdot 197^{3} + 6\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 51 + 31\cdot 197 + 41\cdot 197^{2} + 136\cdot 197^{3} + 120\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 + 115\cdot 197 + 33\cdot 197^{2} + 54\cdot 197^{3} + 65\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 59 + 152\cdot 197 + 74\cdot 197^{2} + 49\cdot 197^{3} + 85\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 106 + 184\cdot 197 + 179\cdot 197^{2} + 75\cdot 197^{3} + 65\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 122 + 68\cdot 197 + 16\cdot 197^{2} + 168\cdot 197^{3} + 177\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 159 + 57\cdot 197 + 72\cdot 197^{2} + 122\cdot 197^{3} + 65\cdot 197^{4} +O(197^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,2)(4,7)(5,8)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,2,3)(4,7,8,5)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,4,6,7,2,8,3,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,7,3,4,2,5,6,8)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |