Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.33973862400.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.15.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{-6})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} + 12x^{4} - 20x^{2} + 25 \)
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The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 54\cdot 107 + 42\cdot 107^{2} + 17\cdot 107^{3} + 53\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 + 58\cdot 107 + 68\cdot 107^{2} + 105\cdot 107^{3} + 24\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 + 74\cdot 107 + 101\cdot 107^{2} + 60\cdot 107^{3} + 107^{4} +O(107^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 87\cdot 107 + 77\cdot 107^{2} + 105\cdot 107^{3} + 53\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 + 19\cdot 107 + 29\cdot 107^{2} + 107^{3} + 53\cdot 107^{4} +O(107^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 64 + 32\cdot 107 + 5\cdot 107^{2} + 46\cdot 107^{3} + 105\cdot 107^{4} +O(107^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 65 + 48\cdot 107 + 38\cdot 107^{2} + 107^{3} + 82\cdot 107^{4} +O(107^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 93 + 52\cdot 107 + 64\cdot 107^{2} + 89\cdot 107^{3} + 53\cdot 107^{4} +O(107^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.