Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1272\)\(\medspace = 2^{3} \cdot 3 \cdot 53 \) |
| Artin stem field: | Galois closure of 8.0.931958784.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.1272.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-6}, \sqrt{106})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 3x^{6} + 42x^{4} - 98x^{3} + 104x^{2} - 68x + 31 \)
|
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 196\cdot 199 + 119\cdot 199^{2} + 191\cdot 199^{3} + 90\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 34\cdot 199 + 179\cdot 199^{2} + 86\cdot 199^{3} + 52\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 51 + 107\cdot 199 + 53\cdot 199^{2} + 18\cdot 199^{3} + 176\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 113 + 60\cdot 199 + 45\cdot 199^{2} + 101\cdot 199^{3} + 78\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 124 + 168\cdot 199 + 78\cdot 199^{2} + 66\cdot 199^{3} + 48\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 133 + 141\cdot 199 + 38\cdot 199^{2} + 98\cdot 199^{3} + 186\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 153 + 134\cdot 199 + 111\cdot 199^{2} + 107\cdot 199^{3} + 183\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 188 + 151\cdot 199 + 168\cdot 199^{2} + 125\cdot 199^{3} + 178\cdot 199^{4} +O(199^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(5,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,3,4,2)(5,6,8,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)(5,7,8,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,4,6)(2,8,3,5)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $0$ |
| $2$ | $4$ | $(1,8,4,5)(2,6,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.