Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.19110297600.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.40.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-6})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 8x^{6} - 12x^{5} + 24x^{4} - 72x^{3} + 116x^{2} - 108x + 97 \)
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The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 80\cdot 131 + 28\cdot 131^{2} + 24\cdot 131^{3} + 120\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 47\cdot 131 + 115\cdot 131^{2} + 13\cdot 131^{3} + 52\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 + 115\cdot 131 + 120\cdot 131^{2} + 34\cdot 131^{3} + 57\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 70 + 46\cdot 131 + 4\cdot 131^{2} + 74\cdot 131^{3} + 63\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 72 + 41\cdot 131 + 117\cdot 131^{2} + 43\cdot 131^{3} + 42\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 89 + 126\cdot 131 + 24\cdot 131^{2} + 130\cdot 131^{3} + 103\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 91 + 8\cdot 131 + 73\cdot 131^{2} + 33\cdot 131^{3} + 105\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 121 + 57\cdot 131 + 39\cdot 131^{2} + 38\cdot 131^{3} + 110\cdot 131^{4} +O(131^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(2,5)(3,8)$ | $0$ |
| $1$ | $4$ | $(1,4,7,6)(2,3,5,8)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,7,4)(2,8,5,3)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,7,6)(2,8,5,3)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,4,5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,7,5)(3,6,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.