Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.0.8192000.1 |
| 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 4.2.2000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{4} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 27\cdot 181 + 151\cdot 181^{2} + 165\cdot 181^{3} + 106\cdot 181^{4} + 35\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 + 22\cdot 181 + 172\cdot 181^{2} + 143\cdot 181^{3} + 44\cdot 181^{4} + 40\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 66 + 37\cdot 181 + 46\cdot 181^{2} + 81\cdot 181^{3} + 159\cdot 181^{4} + 52\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 89 + 26\cdot 181 + 149\cdot 181^{2} + 5\cdot 181^{3} + 90\cdot 181^{4} + 85\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 92 + 154\cdot 181 + 31\cdot 181^{2} + 175\cdot 181^{3} + 90\cdot 181^{4} + 95\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 115 + 143\cdot 181 + 134\cdot 181^{2} + 99\cdot 181^{3} + 21\cdot 181^{4} + 128\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 119 + 158\cdot 181 + 8\cdot 181^{2} + 37\cdot 181^{3} + 136\cdot 181^{4} + 140\cdot 181^{5} +O(181^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 168 + 153\cdot 181 + 29\cdot 181^{2} + 15\cdot 181^{3} + 74\cdot 181^{4} + 145\cdot 181^{5} +O(181^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6,8,3)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,4,7,5)(3,6)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,5,7,4)(3,6)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $0$ |
| $4$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.