Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.38423222208.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.12348.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 42x^{4} - 42x^{3} + 28x^{2} - 24x + 16 \)
|
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 205\cdot 233 + 123\cdot 233^{2} + 161\cdot 233^{3} + 158\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 + 167\cdot 233 + 231\cdot 233^{2} + 53\cdot 233^{3} + 94\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 83 + 16\cdot 233 + 61\cdot 233^{2} + 58\cdot 233^{3} + 80\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 115 + 89\cdot 233 + 174\cdot 233^{2} + 158\cdot 233^{3} + 211\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 146 + 193\cdot 233 + 110\cdot 233^{2} + 110\cdot 233^{3} + 185\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 154 + 101\cdot 233 + 8\cdot 233^{2} + 169\cdot 233^{3} + 19\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 183 + 73\cdot 233^{2} + 138\cdot 233^{3} + 58\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 205 + 157\cdot 233 + 148\cdot 233^{2} + 81\cdot 233^{3} + 123\cdot 233^{4} +O(233^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,2,7)(3,6,5,4)$ | $0$ |
| $2$ | $8$ | $(1,6,8,5,2,4,7,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,7,6,2,3,8,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.