Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3332\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17 \) |
| Artin stem field: | Galois closure of 8.0.8704143616.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.68.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{119})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} + 18x^{5} - 43x^{4} + 4x^{3} + 100x^{2} - 128x + 64 \)
|
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 + 22\cdot 53^{2} + 23\cdot 53^{3} + 26\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 11\cdot 53 + 48\cdot 53^{2} + 27\cdot 53^{3} + 14\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 29\cdot 53 + 9\cdot 53^{2} + 3\cdot 53^{3} + 26\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 + 32\cdot 53 + 16\cdot 53^{2} + 29\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 + 11\cdot 53 + 26\cdot 53^{2} + 10\cdot 53^{3} + 27\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 + 50\cdot 53 + 16\cdot 53^{2} + 49\cdot 53^{3} + 24\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 42 + 9\cdot 53 + 12\cdot 53^{2} + 7\cdot 53^{3} + 41\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 45 + 12\cdot 53 + 7\cdot 53^{2} + 8\cdot 53^{3} + 16\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,4,8,6)(2,5,7,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,4)(2,3,7,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,5)(2,6,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,6,5,4)$ | $0$ |
| $2$ | $4$ | $(1,6,8,4)(2,5,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.