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 8.0.19110297600.5 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} - 12x^{5} + 18x^{4} + 48x^{3} - 16x^{2} - 168x + 166 \)
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The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 29\cdot 59 + 23\cdot 59^{2} + 8\cdot 59^{3} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 58\cdot 59 + 30\cdot 59^{2} + 52\cdot 59^{3} + 29\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 22\cdot 59 + 42\cdot 59^{2} + 36\cdot 59^{3} + 45\cdot 59^{4} +O(59^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 21 + 51\cdot 59 + 49\cdot 59^{2} + 21\cdot 59^{3} + 16\cdot 59^{4} +O(59^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 35 + 49\cdot 59^{2} + 13\cdot 59^{3} + 46\cdot 59^{4} +O(59^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 37 + 37\cdot 59 + 25\cdot 59^{2} + 23\cdot 59^{3} + 19\cdot 59^{4} +O(59^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 45 + 58\cdot 59 + 18\cdot 59^{2} + 5\cdot 59^{3} + 23\cdot 59^{4} +O(59^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 47 + 36\cdot 59 + 54\cdot 59^{2} + 14\cdot 59^{3} + 55\cdot 59^{4} +O(59^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,5,4,8)(2,7,3,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,4,5)(2,6,3,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,4,3)(5,7,8,6)$ | $0$ |
| $2$ | $4$ | $(1,7,4,6)(2,5,3,8)$ | $0$ |
| $2$ | $4$ | $(1,8,4,5)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.