Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(128\)\(\medspace = 2^{7} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.4194304.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 6x^{4} + 1 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 19\cdot 41 + 9\cdot 41^{2} + 18\cdot 41^{3} + 15\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 32\cdot 41 + 15\cdot 41^{2} + 29\cdot 41^{3} + 14\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 11\cdot 41 + 20\cdot 41^{2} + 35\cdot 41^{3} + 35\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 26\cdot 41 + 17\cdot 41^{2} + 39\cdot 41^{3} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 + 14\cdot 41 + 23\cdot 41^{2} + 41^{3} + 40\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 + 29\cdot 41 + 20\cdot 41^{2} + 5\cdot 41^{3} + 5\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 29 + 8\cdot 41 + 25\cdot 41^{2} + 11\cdot 41^{3} + 26\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 30 + 21\cdot 41 + 31\cdot 41^{2} + 22\cdot 41^{3} + 25\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.