Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1680\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Artin stem field: | Galois closure of 8.0.3457440000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.420.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{15}, \sqrt{-21})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 5x^{6} + 14x^{4} - 15x^{2} + 9 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 66\cdot 71 + 30\cdot 71^{2} + 32\cdot 71^{3} + 6\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 62\cdot 71 + 3\cdot 71^{2} + 24\cdot 71^{3} + 11\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 36\cdot 71 + 50\cdot 71^{2} + 21\cdot 71^{3} + 61\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 28\cdot 71 + 14\cdot 71^{2} + 55\cdot 71^{3} + 66\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 52 + 42\cdot 71 + 56\cdot 71^{2} + 15\cdot 71^{3} + 4\cdot 71^{4} +O(71^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 54 + 34\cdot 71 + 20\cdot 71^{2} + 49\cdot 71^{3} + 9\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 61 + 8\cdot 71 + 67\cdot 71^{2} + 46\cdot 71^{3} + 59\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 65 + 4\cdot 71 + 40\cdot 71^{2} + 38\cdot 71^{3} + 64\cdot 71^{4} +O(71^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ | |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |