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 8.0.9032601600.2 |
| 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 \(\Q(\sqrt{-6}, \sqrt{-55})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 7x^{6} + 4x^{5} + 100x^{4} - 262x^{3} + 266x^{2} - 220x + 193 \)
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The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 60 + 81\cdot 127 + 8\cdot 127^{2} + 92\cdot 127^{3} + 71\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 61 + 103\cdot 127 + 117\cdot 127^{2} + 100\cdot 127^{3} + 5\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 63 + 103\cdot 127 + 24\cdot 127^{2} + 11\cdot 127^{3} + 75\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 71 + 92\cdot 127 + 102\cdot 127^{2} + 49\cdot 127^{3} + 101\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 75 + 19\cdot 127 + 66\cdot 127^{2} + 51\cdot 127^{3} + 30\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 88 + 11\cdot 127 + 57\cdot 127^{2} + 80\cdot 127^{3} + 119\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 97 + 69\cdot 127 + 27\cdot 127^{2} + 21\cdot 127^{3} + 122\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 122 + 25\cdot 127 + 103\cdot 127^{2} + 100\cdot 127^{3} + 108\cdot 127^{4} +O(127^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $-2$ | |
| $2$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ | |
| $2$ | $2$ | $(5,6)(7,8)$ | $0$ | |
| $1$ | $4$ | $(1,3,4,2)(5,7,6,8)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,2,4,3)(5,8,6,7)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,3,4,2)(5,8,6,7)$ | $0$ | |
| $2$ | $4$ | $(1,7,4,8)(2,5,3,6)$ | $0$ | |
| $2$ | $4$ | $(1,6,4,5)(2,7,3,8)$ | $0$ |