Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Artin stem field: | Galois closure of 8.0.225815040000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.440.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-30}, \sqrt{-55})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 20x^{6} - 28x^{5} + 67x^{4} - 20x^{3} + 32x^{2} - 8x + 4 \)
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The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 39 + 99\cdot 233 + 116\cdot 233^{2} + 120\cdot 233^{3} + 135\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 + 52\cdot 233 + 88\cdot 233^{2} + 102\cdot 233^{3} + 87\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 114 + 34\cdot 233 + 155\cdot 233^{2} + 80\cdot 233^{3} + 211\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 173 + 117\cdot 233 + 133\cdot 233^{2} + 195\cdot 233^{3} + 62\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 178 + 124\cdot 233 + 166\cdot 233^{2} + 232\cdot 233^{3} + 65\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 189 + 189\cdot 233 + 94\cdot 233^{2} + 10\cdot 233^{3} + 177\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 199 + 3\cdot 233 + 146\cdot 233^{2} + 81\cdot 233^{3} + 212\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 215 + 76\cdot 233 + 31\cdot 233^{2} + 108\cdot 233^{3} + 212\cdot 233^{4} +O(233^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ | |
| $2$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(2,5)(4,8)$ | $0$ | |
| $1$ | $4$ | $(1,3,6,7)(2,4,5,8)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,7,6,3)(2,8,5,4)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,3,6,7)(2,8,5,4)$ | $0$ | |
| $2$ | $4$ | $(1,2,6,5)(3,4,7,8)$ | $0$ | |
| $2$ | $4$ | $(1,4,6,8)(2,3,5,7)$ | $0$ |