Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.2097152000.8 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.8000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{4} + 20 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 31\cdot 61 + 54\cdot 61^{2} + 4\cdot 61^{3} + 49\cdot 61^{4} + 54\cdot 61^{6} + 55\cdot 61^{7} + 25\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 30\cdot 61 + 43\cdot 61^{2} + 7\cdot 61^{3} + 42\cdot 61^{4} + 6\cdot 61^{5} + 2\cdot 61^{6} + 25\cdot 61^{7} + 24\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 25\cdot 61 + 48\cdot 61^{2} + 34\cdot 61^{3} + 29\cdot 61^{4} + 24\cdot 61^{5} + 43\cdot 61^{6} + 29\cdot 61^{7} + 40\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 55\cdot 61 + 25\cdot 61^{2} + 21\cdot 61^{3} + 45\cdot 61^{4} + 20\cdot 61^{5} + 16\cdot 61^{6} + 57\cdot 61^{7} + 60\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 5\cdot 61 + 35\cdot 61^{2} + 39\cdot 61^{3} + 15\cdot 61^{4} + 40\cdot 61^{5} + 44\cdot 61^{6} + 3\cdot 61^{7} +O(61^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 + 35\cdot 61 + 12\cdot 61^{2} + 26\cdot 61^{3} + 31\cdot 61^{4} + 36\cdot 61^{5} + 17\cdot 61^{6} + 31\cdot 61^{7} + 20\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 55 + 30\cdot 61 + 17\cdot 61^{2} + 53\cdot 61^{3} + 18\cdot 61^{4} + 54\cdot 61^{5} + 58\cdot 61^{6} + 35\cdot 61^{7} + 36\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 56 + 29\cdot 61 + 6\cdot 61^{2} + 56\cdot 61^{3} + 11\cdot 61^{4} + 60\cdot 61^{5} + 6\cdot 61^{6} + 5\cdot 61^{7} + 35\cdot 61^{8} +O(61^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,2,8,7)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,5,6,4)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,4,6,5)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $4$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
| $4$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.