Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Artin stem field: | Galois closure of 8.0.5308416.2 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 2x^{6} + 5x^{4} - 4x^{2} + 1 \)
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The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 32\cdot 73 + 55\cdot 73^{2} + 61\cdot 73^{3} + 29\cdot 73^{4} +O(73^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 26 + 58\cdot 73 + 44\cdot 73^{3} + 46\cdot 73^{4} +O(73^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 29 + 41\cdot 73 + 3\cdot 73^{2} + 16\cdot 73^{3} + 38\cdot 73^{4} +O(73^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 36 + 50\cdot 73 + 42\cdot 73^{2} + 41\cdot 73^{3} + 59\cdot 73^{4} +O(73^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 37 + 22\cdot 73 + 30\cdot 73^{2} + 31\cdot 73^{3} + 13\cdot 73^{4} +O(73^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 44 + 31\cdot 73 + 69\cdot 73^{2} + 56\cdot 73^{3} + 34\cdot 73^{4} +O(73^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 47 + 14\cdot 73 + 72\cdot 73^{2} + 28\cdot 73^{3} + 26\cdot 73^{4} +O(73^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 67 + 40\cdot 73 + 17\cdot 73^{2} + 11\cdot 73^{3} + 43\cdot 73^{4} +O(73^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ | |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ | |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |