Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3528\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 8.0.5489031744.1 |
| 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 \(\Q(\sqrt{6}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 16x^{6} + 26x^{5} + 73x^{4} - 176x^{3} + 38x^{2} + 55x + 67 \)
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The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 28\cdot 67 + 15\cdot 67^{3} + 55\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 1 + 45\cdot 67 + 38\cdot 67^{2} + 39\cdot 67^{3} + 50\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 + 64\cdot 67 + 41\cdot 67^{2} + 3\cdot 67^{3} + 8\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 52\cdot 67 + 40\cdot 67^{2} + 9\cdot 67^{3} + 44\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 + 66\cdot 67 + 64\cdot 67^{2} + 43\cdot 67^{3} + 32\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 57 + 55\cdot 67 + 40\cdot 67^{2} + 67^{3} + 9\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 63 + 49\cdot 67 + 34\cdot 67^{2} + 13\cdot 67^{3} + 6\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 64 + 39\cdot 67 + 5\cdot 67^{2} + 7\cdot 67^{3} + 62\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-2$ | |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $0$ | |
| $2$ | $2$ | $(4,7)(6,8)$ | $0$ | |
| $1$ | $4$ | $(1,3,5,2)(4,6,7,8)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,2,5,3)(4,8,7,6)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,4,5,7)(2,8,3,6)$ | $0$ | |
| $2$ | $4$ | $(1,3,5,2)(4,8,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,8,5,6)(2,7,3,4)$ | $0$ |