Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(704\)\(\medspace = 2^{6} \cdot 11 \) |
| Artin stem field: | Galois closure of 8.0.126877696.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.88.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{-11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{6} - 12x^{5} + 12x^{4} + 56x^{3} + 76x^{2} + 52x + 17 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 37\cdot 89 + 62\cdot 89^{2} + 26\cdot 89^{3} + 3\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 67\cdot 89 + 35\cdot 89^{2} + 37\cdot 89^{3} + 4\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 + 28\cdot 89 + 76\cdot 89^{2} + 12\cdot 89^{3} + 60\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 + 34\cdot 89 + 32\cdot 89^{2} + 72\cdot 89^{3} + 8\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 62 + 77\cdot 89 + 6\cdot 89^{2} + 66\cdot 89^{3} + 16\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 66 + 3\cdot 89 + 68\cdot 89^{2} + 87\cdot 89^{3} + 17\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 83 + 44\cdot 89 + 3\cdot 89^{2} + 12\cdot 89^{3} + 21\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 87 + 61\cdot 89 + 70\cdot 89^{2} + 40\cdot 89^{3} + 45\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(2,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,5,3,4)(2,8,7,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,5)(2,6,7,8)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,3,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,3,2)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,3,4)(2,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.