Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.3701505600.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.1560.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-30}, \sqrt{-39})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - 11x^{5} + 45x^{4} - 6x^{3} + 10x^{2} + 36x + 12 \)
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The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 15\cdot 43 + 41\cdot 43^{2} + 42\cdot 43^{3} + 27\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 39\cdot 43 + 27\cdot 43^{2} + 39\cdot 43^{3} + 39\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 8\cdot 43 + 40\cdot 43^{2} + 34\cdot 43^{3} + 12\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 42\cdot 43 + 24\cdot 43^{2} + 9\cdot 43^{3} + 18\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 14\cdot 43 + 23\cdot 43^{2} + 41\cdot 43^{3} + 32\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 + 16\cdot 43 + 36\cdot 43^{2} + 4\cdot 43^{3} + 28\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 27\cdot 43 + 2\cdot 43^{2} + 14\cdot 43^{3} + 38\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 42 + 6\cdot 43 + 18\cdot 43^{2} + 27\cdot 43^{3} + 16\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(3,4)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,5,2,6)(3,8,4,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,2,5)(3,7,4,8)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,2,7)(3,5,4,6)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)(5,7,6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,7,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.