Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(183\)\(\medspace = 3 \cdot 61 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.18385461.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.183.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.549.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 5x^{4} - 9x^{3} + 6x^{2} - 3x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 26\cdot 199 + 82\cdot 199^{2} + 108\cdot 199^{3} + 19\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 48 + 121\cdot 199 + 184\cdot 199^{2} + 111\cdot 199^{3} + 144\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 81 + 58\cdot 199 + 78\cdot 199^{2} + 72\cdot 199^{3} + 38\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 83 + 183\cdot 199 + 194\cdot 199^{2} + 33\cdot 199^{3} + 98\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 99 + 65\cdot 199 + 190\cdot 199^{2} + 111\cdot 199^{3} + 125\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 121 + 125\cdot 199 + 139\cdot 199^{2} + 78\cdot 199^{3} + 156\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 158 + 126\cdot 199 + 63\cdot 199^{2} + 76\cdot 199^{3} + 181\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 184 + 88\cdot 199 + 61\cdot 199^{2} + 3\cdot 199^{3} + 32\cdot 199^{4} +O(199^{5})\)
|
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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,5,3,8)$ | $0$ |
| $2$ | $8$ | $(1,8,6,3,7,5,4,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,4,8,7,2,6,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.