Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Artin stem field: | Galois closure of 8.0.4161798144.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.168.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-6}, \sqrt{-14})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{5} + 24x^{3} + 4x^{2} - 36x + 33 \)
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The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 + 13\cdot 83 + 33\cdot 83^{2} + 71\cdot 83^{3} + 58\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 + 42\cdot 83 + 47\cdot 83^{2} + 58\cdot 83^{3} + 17\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 22\cdot 83 + 30\cdot 83^{2} + 5\cdot 83^{3} + 48\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 + 37\cdot 83 + 54\cdot 83^{2} + 63\cdot 83^{3} + 24\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 + 13\cdot 83 + 10\cdot 83^{2} + 35\cdot 83^{3} + 21\cdot 83^{4} +O(83^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 40 + 17\cdot 83 + 37\cdot 83^{2} + 10\cdot 83^{3} + 55\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 53 + 9\cdot 83 + 71\cdot 83^{2} + 61\cdot 83^{3} + 71\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 67 + 9\cdot 83 + 48\cdot 83^{2} + 25\cdot 83^{3} + 34\cdot 83^{4} +O(83^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,4)(5,7)$ | $-2$ |
| $2$ | $2$ | $(3,4)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,2,8,6)(3,5,4,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,2)(3,7,4,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,6)(3,7,4,5)$ | $0$ |
| $2$ | $4$ | $(1,5,8,7)(2,4,6,3)$ | $0$ |
| $2$ | $4$ | $(1,4,8,3)(2,7,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.