Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(512\)\(\medspace = 2^{9} \) |
| Artin stem field: | Galois closure of 8.2.268435456.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.512.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{4} - 1 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 30\cdot 41 + 11\cdot 41^{2} + 19\cdot 41^{3} + 33\cdot 41^{4} + 30\cdot 41^{5} + 21\cdot 41^{6} + 40\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 38\cdot 41 + 6\cdot 41^{2} + 41^{3} + 40\cdot 41^{5} + 11\cdot 41^{6} + 14\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 5\cdot 41 + 23\cdot 41^{2} + 5\cdot 41^{3} + 3\cdot 41^{4} + 6\cdot 41^{5} + 41^{6} + 37\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 12\cdot 41 + 19\cdot 41^{2} + 26\cdot 41^{3} + 26\cdot 41^{4} + 40\cdot 41^{5} + 18\cdot 41^{6} + 15\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 28\cdot 41 + 21\cdot 41^{2} + 14\cdot 41^{3} + 14\cdot 41^{4} + 22\cdot 41^{6} + 25\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 + 35\cdot 41 + 17\cdot 41^{2} + 35\cdot 41^{3} + 37\cdot 41^{4} + 34\cdot 41^{5} + 39\cdot 41^{6} + 3\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 + 2\cdot 41 + 34\cdot 41^{2} + 39\cdot 41^{3} + 40\cdot 41^{4} + 29\cdot 41^{6} + 26\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 35 + 10\cdot 41 + 29\cdot 41^{2} + 21\cdot 41^{3} + 7\cdot 41^{4} + 10\cdot 41^{5} + 19\cdot 41^{6} +O(41^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.