Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1476\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1429145856.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.164.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.656.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 5x^{6} - 2x^{5} + 21x^{4} - 30x^{3} + 75x^{2} - 62x + 58 \)
|
The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 93\cdot 197 + 77\cdot 197^{2} + 102\cdot 197^{3} + 196\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 + 150\cdot 197 + 15\cdot 197^{2} + 79\cdot 197^{3} + 172\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 + 31\cdot 197 + 175\cdot 197^{2} + 38\cdot 197^{3} + 183\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 101 + 8\cdot 197 + 162\cdot 197^{2} + 139\cdot 197^{3} + 29\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 104 + 28\cdot 197 + 20\cdot 197^{2} + 173\cdot 197^{3} + 103\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 112 + 79\cdot 197 + 82\cdot 197^{2} + 181\cdot 197^{3} + 119\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 161 + 15\cdot 197 + 72\cdot 197^{2} + 167\cdot 197^{3} + 47\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 191 + 183\cdot 197 + 182\cdot 197^{2} + 102\cdot 197^{3} + 131\cdot 197^{4} +O(197^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(3,8)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,6,4,7)(2,3,5,8)$ | $0$ |
| $2$ | $8$ | $(1,3,7,2,4,8,6,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,6,3,4,5,7,8)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.