Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.1024000000.6 |
| 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 \(\Q(i, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} + 16x^{4} - 56x^{2} + 36 \)
|
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 17\cdot 29 + 14\cdot 29^{2} + 29^{3} + 22\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 14\cdot 29 + 19\cdot 29^{2} + 27\cdot 29^{3} + 14\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 5\cdot 29 + 25\cdot 29^{2} + 8\cdot 29^{3} + 19\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 + 2\cdot 29 + 29^{2} + 6\cdot 29^{3} + 12\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 + 26\cdot 29 + 27\cdot 29^{2} + 22\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 23\cdot 29 + 3\cdot 29^{2} + 20\cdot 29^{3} + 9\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 22 + 14\cdot 29 + 9\cdot 29^{2} + 29^{3} + 14\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 27 + 11\cdot 29 + 14\cdot 29^{2} + 27\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,7,5)(2,4,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.