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 8.2.606436875.2 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - x^{6} + 8x^{5} - 2x^{4} - 11x^{3} + 11x^{2} + 13x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 58\cdot 89 + 57\cdot 89^{2} + 73\cdot 89^{3} + 76\cdot 89^{4} + 68\cdot 89^{5} +O(89^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 51\cdot 89 + 63\cdot 89^{2} + 55\cdot 89^{3} + 53\cdot 89^{4} + 63\cdot 89^{5} +O(89^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 32 + 56\cdot 89 + 75\cdot 89^{2} + 44\cdot 89^{3} + 38\cdot 89^{4} + 47\cdot 89^{5} +O(89^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 + 88\cdot 89 + 85\cdot 89^{2} + 77\cdot 89^{3} + 55\cdot 89^{4} + 64\cdot 89^{5} +O(89^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 56\cdot 89 + 67\cdot 89^{2} + 80\cdot 89^{3} + 15\cdot 89^{4} + 26\cdot 89^{5} +O(89^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 67 + 88\cdot 89 + 32\cdot 89^{2} + 69\cdot 89^{3} + 66\cdot 89^{4} + 11\cdot 89^{5} +O(89^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 68 + 48\cdot 89 + 57\cdot 89^{2} + 60\cdot 89^{3} + 25\cdot 89^{4} + 53\cdot 89^{5} +O(89^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 73 + 85\cdot 89 + 3\cdot 89^{2} + 71\cdot 89^{3} + 22\cdot 89^{4} + 20\cdot 89^{5} +O(89^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,5)(4,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(3,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,8,7,4)(2,5,6,3)$ | $0$ |
| $2$ | $8$ | $(1,2,8,5,7,6,4,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,4,2,7,3,8,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.