Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(855\)\(\medspace = 3^{2} \cdot 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1319500125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.95.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 3x^{5} + 9x^{4} + 2x^{3} + 8x^{2} - 4x + 7 \)
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The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 39 + 111\cdot 191 + 175\cdot 191^{2} + 190\cdot 191^{3} + 41\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 54 + 71\cdot 191 + 122\cdot 191^{2} + 175\cdot 191^{3} + 108\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 100 + 63\cdot 191 + 99\cdot 191^{2} + 63\cdot 191^{3} + 190\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 114 + 149\cdot 191 + 64\cdot 191^{2} + 116\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 139 + 2\cdot 191 + 128\cdot 191^{2} + 60\cdot 191^{3} + 54\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 160 + 97\cdot 191 + 7\cdot 191^{2} + 97\cdot 191^{3} + 135\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 163 + 9\cdot 191 + 105\cdot 191^{2} + 4\cdot 191^{3} + 101\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 187 + 66\cdot 191 + 61\cdot 191^{2} + 171\cdot 191^{3} + 15\cdot 191^{4} +O(191^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,4)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,4,2,5)(3,8,7,6)$ | $0$ |
| $2$ | $8$ | $(1,8,4,7,2,6,5,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,5,8,2,3,4,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.