Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1258815488.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1568.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} + 5x^{4} + 2x^{2} + 2 \)
|
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 28\cdot 191 + 2\cdot 191^{2} + 182\cdot 191^{3} + 57\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 58 + 155\cdot 191 + 16\cdot 191^{2} + 65\cdot 191^{3} + 76\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 60 + 38\cdot 191 + 128\cdot 191^{2} + 36\cdot 191^{3} + 117\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 72 + 98\cdot 191 + 121\cdot 191^{2} + 20\cdot 191^{3} + 150\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 119 + 92\cdot 191 + 69\cdot 191^{2} + 170\cdot 191^{3} + 40\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 131 + 152\cdot 191 + 62\cdot 191^{2} + 154\cdot 191^{3} + 73\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 133 + 35\cdot 191 + 174\cdot 191^{2} + 125\cdot 191^{3} + 114\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 151 + 162\cdot 191 + 188\cdot 191^{2} + 8\cdot 191^{3} + 133\cdot 191^{4} +O(191^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.