Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.207360000.5 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} - 6x^{5} + 14x^{4} - 30x^{3} + 50x^{2} - 50x + 25 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 2\cdot 41 + 38\cdot 41^{2} + 6\cdot 41^{3} + 13\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 29\cdot 41 + 37\cdot 41^{2} + 13\cdot 41^{3} + 21\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 3\cdot 41 + 24\cdot 41^{2} + 14\cdot 41^{3} + 27\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 7\cdot 41 + 34\cdot 41^{2} + 34\cdot 41^{3} + 30\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 37\cdot 41 + 29\cdot 41^{2} + 23\cdot 41^{3} + 36\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 3\cdot 41 + 23\cdot 41^{2} + 29\cdot 41^{3} + 18\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 + 24\cdot 41 + 12\cdot 41^{2} + 9\cdot 41^{3} + 39\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 36 + 13\cdot 41 + 5\cdot 41^{2} + 31\cdot 41^{3} + 17\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,3,6)(2,7,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.