Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(471\)\(\medspace = 3 \cdot 157 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.313461333.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.471.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1413.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 4x^{6} + x^{4} + 3x^{3} - 3x^{2} + 9x + 9 \)
|
The roots of $f$ are computed in $\Q_{ 349 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 60\cdot 349 + 46\cdot 349^{2} + 52\cdot 349^{3} + 125\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 191\cdot 349 + 196\cdot 349^{2} + 72\cdot 349^{3} + 151\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 68 + 136\cdot 349 + 52\cdot 349^{2} + 88\cdot 349^{3} + 129\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 71 + 91\cdot 349 + 150\cdot 349^{2} + 124\cdot 349^{3} + 177\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 132 + 269\cdot 349 + 346\cdot 349^{2} + 146\cdot 349^{3} + 131\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 162 + 302\cdot 349 + 338\cdot 349^{2} + 327\cdot 349^{3} + 184\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 243 + 239\cdot 349 + 229\cdot 349^{2} + 17\cdot 349^{3} + 348\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 331 + 105\cdot 349 + 35\cdot 349^{2} + 217\cdot 349^{3} + 148\cdot 349^{4} +O(349^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,8,4,6)(2,7,5,3)$ | $0$ |
| $2$ | $8$ | $(1,2,6,3,4,5,8,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,8,2,4,7,6,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.