Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.47775744.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\zeta_{12})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 3x^{4} + 9 \)
|
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 41\cdot 109 + 28\cdot 109^{2} + 42\cdot 109^{3} + 7\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 10\cdot 109 + 34\cdot 109^{2} + 21\cdot 109^{3} + 30\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 40 + 84\cdot 109 + 14\cdot 109^{2} + 91\cdot 109^{3} + 49\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 53 + 109^{2} + 76\cdot 109^{3} + 49\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 + 108\cdot 109 + 107\cdot 109^{2} + 32\cdot 109^{3} + 59\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 69 + 24\cdot 109 + 94\cdot 109^{2} + 17\cdot 109^{3} + 59\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 97 + 98\cdot 109 + 74\cdot 109^{2} + 87\cdot 109^{3} + 78\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 104 + 67\cdot 109 + 80\cdot 109^{2} + 66\cdot 109^{3} + 101\cdot 109^{4} +O(109^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.