Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1700\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 17 \) |
| Artin stem field: | Galois closure of 8.0.1156000000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.340.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{6} + 16x^{4} - 10x^{3} - 16x^{2} + 50x + 41 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 57\cdot 89 + 35\cdot 89^{2} + 18\cdot 89^{3} + 13\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 84\cdot 89 + 53\cdot 89^{2} + 21\cdot 89^{3} + 27\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 + 25\cdot 89 + 59\cdot 89^{2} + 34\cdot 89^{3} + 42\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 + 83\cdot 89 + 47\cdot 89^{2} + 76\cdot 89^{3} + 46\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 12\cdot 89 + 51\cdot 89^{2} + 78\cdot 89^{3} + 4\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 + 70\cdot 89 + 82\cdot 89^{2} + 87\cdot 89^{3} + 77\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 37 + 29\cdot 89 + 82\cdot 89^{2} + 80\cdot 89^{3} + 25\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 62 + 83\cdot 89 + 31\cdot 89^{2} + 46\cdot 89^{3} + 28\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,2,5,6)(3,7,8,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,5,2)(3,4,8,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,5,4)(2,8,6,3)$ | $0$ |
| $2$ | $4$ | $(1,8,5,3)(2,4,6,7)$ | $0$ |
| $2$ | $4$ | $(1,2,5,6)(3,4,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.