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 8.0.468512000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.220.4t1.a.b |
| 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 $\Q_{ 199 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 164\cdot 199 + 89\cdot 199^{2} + 26\cdot 199^{3} + 89\cdot 199^{4} + 182\cdot 199^{5} + 182\cdot 199^{6} + 125\cdot 199^{7} +O(199^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 50 + 46\cdot 199 + 169\cdot 199^{2} + 102\cdot 199^{3} + 128\cdot 199^{4} + 169\cdot 199^{5} + 3\cdot 199^{6} + 51\cdot 199^{7} +O(199^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 52 + 100\cdot 199 + 161\cdot 199^{2} + 73\cdot 199^{3} + 26\cdot 199^{4} + 189\cdot 199^{5} + 130\cdot 199^{6} + 150\cdot 199^{7} +O(199^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 71 + 168\cdot 199 + 122\cdot 199^{2} + 149\cdot 199^{3} + 46\cdot 199^{4} + 86\cdot 199^{5} + 80\cdot 199^{6} + 124\cdot 199^{7} +O(199^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 128 + 30\cdot 199 + 76\cdot 199^{2} + 49\cdot 199^{3} + 152\cdot 199^{4} + 112\cdot 199^{5} + 118\cdot 199^{6} + 74\cdot 199^{7} +O(199^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 147 + 98\cdot 199 + 37\cdot 199^{2} + 125\cdot 199^{3} + 172\cdot 199^{4} + 9\cdot 199^{5} + 68\cdot 199^{6} + 48\cdot 199^{7} +O(199^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 149 + 152\cdot 199 + 29\cdot 199^{2} + 96\cdot 199^{3} + 70\cdot 199^{4} + 29\cdot 199^{5} + 195\cdot 199^{6} + 147\cdot 199^{7} +O(199^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 189 + 34\cdot 199 + 109\cdot 199^{2} + 172\cdot 199^{3} + 109\cdot 199^{4} + 16\cdot 199^{5} + 16\cdot 199^{6} + 73\cdot 199^{7} +O(199^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(2,3,7,6)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,6,7,3)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,3,7,6)(4,5)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $0$ |
| $4$ | $8$ | $(1,3,4,7,8,6,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.