Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1575\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7 \) |
| Artin stem field: | Galois closure of 8.0.558140625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.35.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 10x^{6} + 11x^{5} + 34x^{4} - 29x^{3} - 55x^{2} + 4x + 76 \)
|
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 82\cdot 109 + 67\cdot 109^{2} + 26\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 48\cdot 109 + 52\cdot 109^{2} + 88\cdot 109^{3} + 101\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 + 97\cdot 109 + 26\cdot 109^{2} + 84\cdot 109^{3} + 23\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 40 + 108\cdot 109 + 14\cdot 109^{2} + 80\cdot 109^{3} + 55\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 48 + 54\cdot 109 + 82\cdot 109^{2} + 7\cdot 109^{3} + 101\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 83 + 92\cdot 109 + 99\cdot 109^{2} + 3\cdot 109^{3} + 65\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 89 + 6\cdot 109 + 6\cdot 109^{2} + 68\cdot 109^{3} + 35\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 90 + 54\cdot 109 + 85\cdot 109^{2} + 102\cdot 109^{3} + 26\cdot 109^{4} +O(109^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(3,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,4,2,6)(3,7,5,8)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,2,4)(3,8,5,7)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,2,8)(3,4,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,3)(4,8,6,7)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.