Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Artin stem field: | Galois closure of 8.0.321978368.5 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.68.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.19652.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 2x^{4} + 17 \)
|
The roots of $f$ are computed in $\Q_{ 293 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 183\cdot 293 + 196\cdot 293^{2} + 211\cdot 293^{3} + 193\cdot 293^{4} + 172\cdot 293^{5} + 152\cdot 293^{6} +O(293^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 119 + 142\cdot 293 + 236\cdot 293^{2} + 139\cdot 293^{3} + 96\cdot 293^{4} + 67\cdot 293^{5} + 17\cdot 293^{6} +O(293^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 122 + 293 + 222\cdot 293^{2} + 269\cdot 293^{3} + 115\cdot 293^{4} + 173\cdot 293^{5} + 7\cdot 293^{6} +O(293^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 135 + 41\cdot 293 + 35\cdot 293^{2} + 103\cdot 293^{3} + 63\cdot 293^{4} + 63\cdot 293^{5} + 183\cdot 293^{6} +O(293^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 158 + 251\cdot 293 + 257\cdot 293^{2} + 189\cdot 293^{3} + 229\cdot 293^{4} + 229\cdot 293^{5} + 109\cdot 293^{6} +O(293^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 171 + 291\cdot 293 + 70\cdot 293^{2} + 23\cdot 293^{3} + 177\cdot 293^{4} + 119\cdot 293^{5} + 285\cdot 293^{6} +O(293^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 174 + 150\cdot 293 + 56\cdot 293^{2} + 153\cdot 293^{3} + 196\cdot 293^{4} + 225\cdot 293^{5} + 275\cdot 293^{6} +O(293^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 279 + 109\cdot 293 + 96\cdot 293^{2} + 81\cdot 293^{3} + 99\cdot 293^{4} + 120\cdot 293^{5} + 140\cdot 293^{6} +O(293^{7})\)
|
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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,2,8,7)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,5,6,4)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,4,6,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $0$ |
| $4$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.