Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(495\)\(\medspace = 3^{2} \cdot 5 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1334161125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.55.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.2475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 3x^{6} - 7x^{5} + 10x^{4} - 6x^{3} + 15x^{2} - 9x + 3 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 9\cdot 89 + 13\cdot 89^{2} + 37\cdot 89^{3} + 32\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 8\cdot 89 + 2\cdot 89^{2} + 23\cdot 89^{3} + 72\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 70\cdot 89 + 20\cdot 89^{2} + 36\cdot 89^{3} + 68\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 53 + 3\cdot 89 + 9\cdot 89^{2} + 56\cdot 89^{3} + 34\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 + 56\cdot 89 + 65\cdot 89^{2} + 25\cdot 89^{3} + 79\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 + 52\cdot 89 + 81\cdot 89^{2} + 15\cdot 89^{3} + 29\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 69 + 72\cdot 89 + 56\cdot 89^{2} + 83\cdot 89^{3} + 61\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 79 + 82\cdot 89 + 17\cdot 89^{2} + 78\cdot 89^{3} + 66\cdot 89^{4} +O(89^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,6,2)(3,8,4,7)$ | $0$ |
| $2$ | $8$ | $(1,3,5,8,6,4,2,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,2,3,6,7,5,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.