Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 8.0.1372257936.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{3}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 7x^{6} + 7x^{5} + 28x^{4} - 7x^{3} - 7x^{2} - 79x + 67 \)
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The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 101\cdot 109 + 100\cdot 109^{2} + 45\cdot 109^{3} + 23\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 100\cdot 109 + 46\cdot 109^{2} + 30\cdot 109^{3} + 34\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 92\cdot 109 + 35\cdot 109^{2} + 70\cdot 109^{3} + 96\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 73\cdot 109 + 67\cdot 109^{2} + 86\cdot 109^{3} + 39\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 45\cdot 109 + 8\cdot 109^{2} + 26\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 56 + 61\cdot 109 + 67\cdot 109^{2} + 30\cdot 109^{3} + 47\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 74 + 33\cdot 109 + 34\cdot 109^{2} + 71\cdot 109^{3} + 63\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 96 + 37\cdot 109 + 74\cdot 109^{2} + 74\cdot 109^{3} + 24\cdot 109^{4} +O(109^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,6,7,4)(2,8,3,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,7,6)(2,5,3,8)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,7,3)(4,5,6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,8,3,5)$ | $0$ |
| $2$ | $4$ | $(1,5,7,8)(2,4,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.