Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Artin stem field: | Galois closure of 8.0.3057647616.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\zeta_{12})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} + 54x^{4} - 72x^{2} + 36 \)
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The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 53\cdot 73 + 59\cdot 73^{2} + 20\cdot 73^{3} + 61\cdot 73^{4} + 37\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 52\cdot 73 + 71\cdot 73^{2} + 31\cdot 73^{3} + 15\cdot 73^{4} + 29\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 53\cdot 73 + 69\cdot 73^{2} + 16\cdot 73^{3} + 57\cdot 73^{4} + 40\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 + 19\cdot 73 + 3\cdot 73^{2} + 24\cdot 73^{3} + 17\cdot 73^{4} + 41\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 53\cdot 73 + 69\cdot 73^{2} + 48\cdot 73^{3} + 55\cdot 73^{4} + 31\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 49 + 19\cdot 73 + 3\cdot 73^{2} + 56\cdot 73^{3} + 15\cdot 73^{4} + 32\cdot 73^{5} +O(73^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 62 + 20\cdot 73 + 73^{2} + 41\cdot 73^{3} + 57\cdot 73^{4} + 43\cdot 73^{5} +O(73^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 65 + 19\cdot 73 + 13\cdot 73^{2} + 52\cdot 73^{3} + 11\cdot 73^{4} + 35\cdot 73^{5} +O(73^{6})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ | |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ | |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |