Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(1089\)\(\medspace = 3^{2} \cdot 11^{2} \) |
| Artin stem field: | Galois closure of 8.2.14206147659.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.3993.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - x^{6} - 5x^{5} + 16x^{4} - x^{3} - 4x^{2} - 28x + 16 \)
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The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 162\cdot 163 + 150\cdot 163^{2} + 20\cdot 163^{3} + 10\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 134\cdot 163 + 120\cdot 163^{2} + 24\cdot 163^{3} + 62\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 85\cdot 163 + 113\cdot 163^{2} + 108\cdot 163^{3} + 147\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 + 72\cdot 163 + 48\cdot 163^{2} + 155\cdot 163^{3} + 119\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 + 115\cdot 163 + 21\cdot 163^{2} + 157\cdot 163^{3} + 31\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 72 + 108\cdot 163 + 23\cdot 163^{2} + 104\cdot 163^{3} + 44\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 109 + 107\cdot 163 + 103\cdot 163^{2} + 8\cdot 163^{3} + 106\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 145 + 29\cdot 163 + 69\cdot 163^{2} + 72\cdot 163^{3} + 129\cdot 163^{4} +O(163^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,7)(4,6,5,8)$ | $0$ |
| $4$ | $4$ | $(1,5,2,4)(3,6,7,8)$ | $0$ |
| $2$ | $8$ | $(1,4,3,6,2,5,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,8,2,4,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.