Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(184\)\(\medspace = 2^{3} \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.1146228736.5 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.184.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-23})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - 4x^{5} + 43x^{4} + 104x^{3} + 86x^{2} + 24x + 36 \)
|
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 41\cdot 47 + 31\cdot 47^{2} + 28\cdot 47^{3} + 38\cdot 47^{4} + 43\cdot 47^{5} + 13\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 16\cdot 47 + 11\cdot 47^{2} + 27\cdot 47^{3} + 4\cdot 47^{4} + 17\cdot 47^{5} + 25\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 27\cdot 47 + 4\cdot 47^{2} + 32\cdot 47^{3} + 27\cdot 47^{4} + 14\cdot 47^{5} + 12\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 5\cdot 47 + 45\cdot 47^{2} + 21\cdot 47^{3} + 35\cdot 47^{4} + 13\cdot 47^{5} + 23\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 31\cdot 47 + 29\cdot 47^{2} + 15\cdot 47^{3} + 27\cdot 47^{5} + 22\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 5\cdot 47 + 21\cdot 47^{2} + 22\cdot 47^{3} + 3\cdot 47^{4} + 6\cdot 47^{5} + 32\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 + 9\cdot 47 + 46\cdot 47^{2} + 5\cdot 47^{3} + 23\cdot 47^{4} + 18\cdot 47^{5} + 42\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 45 + 4\cdot 47 + 45\cdot 47^{2} + 33\cdot 47^{3} + 7\cdot 47^{4} + 16\cdot 47^{6} +O(47^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.