Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Artin stem field: | Galois closure of 8.0.9437184.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.24.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{-3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} - 4x^{5} + 6x^{4} + 16x^{3} + 16x^{2} + 8x + 2 \)
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The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 13\cdot 73 + 29\cdot 73^{2} + 30\cdot 73^{3} + 13\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 70\cdot 73 + 13\cdot 73^{2} + 5\cdot 73^{3} + 8\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 37\cdot 73 + 61\cdot 73^{2} + 3\cdot 73^{3} + 21\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 10\cdot 73 + 14\cdot 73^{2} + 49\cdot 73^{3} + 13\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 + 28\cdot 73 + 56\cdot 73^{2} + 14\cdot 73^{3} + 30\cdot 73^{4} +O(73^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 29 + 25\cdot 73 + 41\cdot 73^{2} + 33\cdot 73^{3} + 30\cdot 73^{4} +O(73^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 34 + 37\cdot 73 + 32\cdot 73^{2} + 53\cdot 73^{3} + 55\cdot 73^{4} +O(73^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 71 + 69\cdot 73 + 42\cdot 73^{2} + 28\cdot 73^{3} + 46\cdot 73^{4} +O(73^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $-2$ | |
| $2$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,6)(7,8)$ | $0$ | |
| $2$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ | |
| $1$ | $4$ | $(1,7,6,8)(2,5,3,4)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,8,6,7)(2,4,3,5)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,2,6,3)(4,7,5,8)$ | $0$ | |
| $2$ | $4$ | $(1,4,6,5)(2,8,3,7)$ | $0$ | |
| $2$ | $4$ | $(1,8,6,7)(2,5,3,4)$ | $0$ |