Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Artin stem field: | Galois closure of 8.0.260112384.10 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.168.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-14})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{6} + 26x^{4} - 70x^{2} + 49 \)
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The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 56\cdot 61 + 25\cdot 61^{2} + 38\cdot 61^{3} + 7\cdot 61^{4} + 19\cdot 61^{5} +O(61^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 12 + 40\cdot 61 + 24\cdot 61^{2} + 36\cdot 61^{3} + 36\cdot 61^{4} + 36\cdot 61^{5} +O(61^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 20 + 58\cdot 61 + 46\cdot 61^{2} + 46\cdot 61^{3} + 46\cdot 61^{4} + 61^{5} +O(61^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 29 + 14\cdot 61 + 48\cdot 61^{2} + 4\cdot 61^{3} + 19\cdot 61^{4} + 58\cdot 61^{5} +O(61^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 32 + 46\cdot 61 + 12\cdot 61^{2} + 56\cdot 61^{3} + 41\cdot 61^{4} + 2\cdot 61^{5} +O(61^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 41 + 2\cdot 61 + 14\cdot 61^{2} + 14\cdot 61^{3} + 14\cdot 61^{4} + 59\cdot 61^{5} +O(61^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 49 + 20\cdot 61 + 36\cdot 61^{2} + 24\cdot 61^{3} + 24\cdot 61^{4} + 24\cdot 61^{5} +O(61^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 52 + 4\cdot 61 + 35\cdot 61^{2} + 22\cdot 61^{3} + 53\cdot 61^{4} + 41\cdot 61^{5} +O(61^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.