Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(3328\)\(\medspace = 2^{8} \cdot 13 \) |
| Artin number field: | Galois closure of 8.0.36859543552.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.35152.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 60\cdot 181 + 93\cdot 181^{2} + 54\cdot 181^{3} + 136\cdot 181^{4} + 148\cdot 181^{5} + 114\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 36\cdot 181 + 129\cdot 181^{2} + 161\cdot 181^{3} + 63\cdot 181^{4} + 95\cdot 181^{5} + 153\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 + 9\cdot 181 + 134\cdot 181^{2} + 66\cdot 181^{3} + 159\cdot 181^{4} + 103\cdot 181^{5} + 5\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 + 21\cdot 181 + 6\cdot 181^{2} + 15\cdot 181^{3} + 89\cdot 181^{4} + 46\cdot 181^{5} + 20\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 104 + 159\cdot 181 + 174\cdot 181^{2} + 165\cdot 181^{3} + 91\cdot 181^{4} + 134\cdot 181^{5} + 160\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 135 + 171\cdot 181 + 46\cdot 181^{2} + 114\cdot 181^{3} + 21\cdot 181^{4} + 77\cdot 181^{5} + 175\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 150 + 144\cdot 181 + 51\cdot 181^{2} + 19\cdot 181^{3} + 117\cdot 181^{4} + 85\cdot 181^{5} + 27\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 166 + 120\cdot 181 + 87\cdot 181^{2} + 126\cdot 181^{3} + 44\cdot 181^{4} + 32\cdot 181^{5} + 66\cdot 181^{6} +O(181^{7})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,5,8,4)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,3,7,6)(4,5)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $0$ | $0$ |