Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(183\)\(\medspace = 3 \cdot 61 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.1121513121.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.183.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{61})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 7x^{6} + 52x^{4} - 21x^{2} + 9 \)
|
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 73 + 29\cdot 73^{2} + 34\cdot 73^{3} + 58\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 4\cdot 73 + 39\cdot 73^{2} + 67\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 53\cdot 73 + 13\cdot 73^{2} + 9\cdot 73^{3} + 19\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 + 36\cdot 73 + 66\cdot 73^{2} + 5\cdot 73^{3} + 40\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 + 36\cdot 73 + 6\cdot 73^{2} + 67\cdot 73^{3} + 32\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 + 19\cdot 73 + 59\cdot 73^{2} + 63\cdot 73^{3} + 53\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 50 + 68\cdot 73 + 33\cdot 73^{2} + 5\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 61 + 71\cdot 73 + 43\cdot 73^{2} + 38\cdot 73^{3} + 14\cdot 73^{4} +O(73^{5})\)
|
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.