Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(299\)\(\medspace = 13 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.347501687.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.299.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.3887.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 5x^{6} + 6x^{5} + 2x^{4} - 10x^{3} - 7x^{2} - x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 36\cdot 131 + 66\cdot 131^{2} + 131^{3} + 76\cdot 131^{4} + 87\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 43 + 30\cdot 131 + 60\cdot 131^{2} + 49\cdot 131^{3} + 41\cdot 131^{4} + 36\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 56 + 83\cdot 131 + 107\cdot 131^{2} + 99\cdot 131^{3} + 65\cdot 131^{4} + 112\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 + 75\cdot 131 + 95\cdot 131^{2} + 71\cdot 131^{3} + 18\cdot 131^{4} + 57\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 62 + 24\cdot 131 + 112\cdot 131^{2} + 74\cdot 131^{3} + 67\cdot 131^{4} + 93\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 80 + 100\cdot 131 + 97\cdot 131^{2} + 129\cdot 131^{3} + 80\cdot 131^{4} + 103\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 91 + 49\cdot 131 + 77\cdot 131^{2} + 13\cdot 131^{3} + 71\cdot 131^{4} + 131^{5} +O(131^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 107 + 123\cdot 131 + 37\cdot 131^{2} + 83\cdot 131^{3} + 102\cdot 131^{4} + 31\cdot 131^{5} +O(131^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(2,4)(3,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,5,2)(3,7,6,8)$ | $0$ |
| $2$ | $8$ | $(1,8,4,3,5,7,2,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,8,5,6,4,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.