Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3920\)\(\medspace = 2^{4} \cdot 5 \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 8.0.12047257600.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{35})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 10x^{5} + 19x^{4} + 2x^{3} - 40x^{2} + 24x + 74 \)
|
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 70\cdot 109 + 98\cdot 109^{2} + 57\cdot 109^{3} + 46\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 104\cdot 109^{2} + 41\cdot 109^{3} + 49\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 + 56\cdot 109 + 99\cdot 109^{2} + 35\cdot 109^{3} + 41\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 77\cdot 109 + 75\cdot 109^{2} + 31\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 44 + 84\cdot 109 + 47\cdot 109^{2} + 108\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 + 45\cdot 109 + 19\cdot 109^{2} + 32\cdot 109^{3} + 70\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 75 + 91\cdot 109 + 24\cdot 109^{2} + 82\cdot 109^{3} + 80\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 90 + 10\cdot 109 + 75\cdot 109^{2} + 45\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,4,7,5)(2,6,3,8)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,7,4)(2,8,3,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,7,3)(4,6,5,8)$ | $0$ |
| $2$ | $4$ | $(1,8,7,6)(2,5,3,4)$ | $0$ |
| $2$ | $4$ | $(1,5,7,4)(2,6,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.