Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Artin stem field: | Galois closure of 8.0.70560000.2 |
| 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 \(\Q(\sqrt{-5}, \sqrt{-21})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{6} - 4x^{5} + 4x^{4} + 2x^{3} + 4x^{2} - 10x + 5 \)
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The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 30\cdot 101 + 100\cdot 101^{2} + 100\cdot 101^{3} + 11\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 84\cdot 101 + 84\cdot 101^{2} + 29\cdot 101^{3} + 14\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 52 + 100\cdot 101 + 29\cdot 101^{2} + 59\cdot 101^{3} + 94\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 + 87\cdot 101 + 78\cdot 101^{2} + 68\cdot 101^{3} + 54\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 66 + 84\cdot 101 + 93\cdot 101^{2} + 73\cdot 101^{3} + 40\cdot 101^{4} +O(101^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 89 + 57\cdot 101 + 74\cdot 101^{2} + 78\cdot 101^{3} + 79\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 94 + 82\cdot 101 + 58\cdot 101^{2} + 46\cdot 101^{3} + 70\cdot 101^{4} +O(101^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 95 + 77\cdot 101 + 84\cdot 101^{2} + 46\cdot 101^{3} + 37\cdot 101^{4} +O(101^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,8,3,6)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,3,8)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,3,2)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,6,3,8)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,4,3,5)(2,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.