Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2040\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 17 \) |
| Artin stem field: | Galois closure of 8.0.59927040000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.2040.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-30}, \sqrt{34})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 3x^{6} + 10x^{5} - 14x^{3} + 125x^{2} + 6x + 1 \)
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The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 143\cdot 151 + 115\cdot 151^{2} + 93\cdot 151^{3} + 74\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 93\cdot 151 + 105\cdot 151^{2} + 126\cdot 151^{3} + 85\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 71 + 66\cdot 151 + 54\cdot 151^{2} + 145\cdot 151^{3} + 30\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 72 + 83\cdot 151 + 59\cdot 151^{2} + 107\cdot 151^{3} + 133\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 103 + 54\cdot 151 + 113\cdot 151^{2} + 120\cdot 151^{3} + 150\cdot 151^{4} +O(151^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 104 + 70\cdot 151 + 23\cdot 151^{2} + 98\cdot 151^{3} + 82\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 105 + 87\cdot 151 + 28\cdot 151^{2} + 60\cdot 151^{3} + 34\cdot 151^{4} +O(151^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 108 + 4\cdot 151 + 103\cdot 151^{2} + 2\cdot 151^{3} + 11\cdot 151^{4} +O(151^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ | |
| $2$ | $2$ | $(2,5)(4,6)$ | $0$ | |
| $2$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $0$ | ✓ |
| $1$ | $4$ | $(1,7,8,3)(2,4,5,6)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,3,8,7)(2,6,5,4)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,6,8,4)(2,3,5,7)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,3)(2,6,5,4)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,5)(3,6,7,4)$ | $0$ |