Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(644\)\(\medspace = 2^{2} \cdot 7 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1869629888.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.644.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.4508.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 4x^{6} + 3x^{5} + 3x^{4} - 12x^{3} - 6x^{2} + 8x + 4 \)
|
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 22\cdot 43 + 27\cdot 43^{2} + 13\cdot 43^{3} + 9\cdot 43^{4} + 16\cdot 43^{5} + 43^{6} +O(43^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 43 + 25\cdot 43^{2} + 29\cdot 43^{3} + 20\cdot 43^{4} + 20\cdot 43^{5} + 31\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 17\cdot 43 + 27\cdot 43^{2} + 3\cdot 43^{3} + 19\cdot 43^{4} + 8\cdot 43^{5} + 32\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 9\cdot 43 + 5\cdot 43^{2} + 25\cdot 43^{3} + 26\cdot 43^{4} + 5\cdot 43^{5} + 36\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 43 + 39\cdot 43^{2} + 27\cdot 43^{3} + 8\cdot 43^{4} + 41\cdot 43^{5} + 8\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 7\cdot 43 + 36\cdot 43^{2} + 37\cdot 43^{3} + 36\cdot 43^{4} + 36\cdot 43^{5} + 10\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 29 + 37\cdot 43 + 7\cdot 43^{2} + 14\cdot 43^{3} + 13\cdot 43^{4} + 43^{5} + 6\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 36 + 32\cdot 43 + 3\cdot 43^{2} + 20\cdot 43^{3} + 37\cdot 43^{4} + 41\cdot 43^{5} + 43^{6} +O(43^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,8)$ | $0$ |
| $2$ | $4$ | $(1,5,3,8)(2,4,6,7)$ | $0$ |
| $2$ | $8$ | $(1,4,5,6,3,7,8,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,6,8,4,3,2,5,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.