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 8.2.373837707.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 2x^{6} - 7x^{5} + 10x^{4} + 7x^{3} - 12x^{2} - x + 5 \)
|
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 49\cdot 199 + 49\cdot 199^{2} + 188\cdot 199^{3} + 126\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 191\cdot 199 + 130\cdot 199^{2} + 190\cdot 199^{3} + 82\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 68 + 73\cdot 199 + 57\cdot 199^{2} + 37\cdot 199^{3} + 130\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 103 + 84\cdot 199 + 160\cdot 199^{2} + 180\cdot 199^{3} + 57\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 121 + 91\cdot 199 + 73\cdot 199^{2} + 21\cdot 199^{3} + 145\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 143 + 16\cdot 199 + 125\cdot 199^{2} + 79\cdot 199^{3} + 14\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 164 + 42\cdot 199 + 74\cdot 199^{2} + 106\cdot 199^{3} + 136\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 169 + 47\cdot 199 + 125\cdot 199^{2} + 190\cdot 199^{3} + 101\cdot 199^{4} +O(199^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,4)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)(5,8,6,7)$ | $0$ |
| $2$ | $8$ | $(1,6,3,7,4,5,2,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,2,6,4,8,3,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.