Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| Artin stem field: | Galois closure of 8.0.468512000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.220.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.22000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{6} + 8x^{4} - 10x^{2} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 50\cdot 59 + 37\cdot 59^{2} + 5\cdot 59^{3} + 11\cdot 59^{4} + 14\cdot 59^{5} + 8\cdot 59^{6} + 58\cdot 59^{7} + 4\cdot 59^{8} +O(59^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 16\cdot 59 + 34\cdot 59^{2} + 22\cdot 59^{3} + 47\cdot 59^{4} + 54\cdot 59^{5} + 17\cdot 59^{6} + 9\cdot 59^{7} + 56\cdot 59^{8} +O(59^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 48\cdot 59 + 11\cdot 59^{2} + 54\cdot 59^{3} + 40\cdot 59^{4} + 3\cdot 59^{5} + 11\cdot 59^{6} + 30\cdot 59^{7} + 31\cdot 59^{8} +O(59^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 31\cdot 59 + 17\cdot 59^{2} + 30\cdot 59^{3} + 54\cdot 59^{4} + 27\cdot 59^{5} + 45\cdot 59^{6} + 41\cdot 59^{7} + 23\cdot 59^{8} +O(59^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 + 27\cdot 59 + 41\cdot 59^{2} + 28\cdot 59^{3} + 4\cdot 59^{4} + 31\cdot 59^{5} + 13\cdot 59^{6} + 17\cdot 59^{7} + 35\cdot 59^{8} +O(59^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 39 + 10\cdot 59 + 47\cdot 59^{2} + 4\cdot 59^{3} + 18\cdot 59^{4} + 55\cdot 59^{5} + 47\cdot 59^{6} + 28\cdot 59^{7} + 27\cdot 59^{8} +O(59^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 47 + 42\cdot 59 + 24\cdot 59^{2} + 36\cdot 59^{3} + 11\cdot 59^{4} + 4\cdot 59^{5} + 41\cdot 59^{6} + 49\cdot 59^{7} + 2\cdot 59^{8} +O(59^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 50 + 8\cdot 59 + 21\cdot 59^{2} + 53\cdot 59^{3} + 47\cdot 59^{4} + 44\cdot 59^{5} + 50\cdot 59^{6} + 54\cdot 59^{8} +O(59^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,3)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,3)(2,7)(4,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3,8,6)(2,7)(4,5)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $0$ |
| $4$ | $8$ | $(1,7,6,5,8,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.