Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(6400\)\(\medspace = 2^{8} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.0.2097152000.7 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.b |
| 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 $\Q_{ 421 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 113\cdot 421 + 277\cdot 421^{2} + 130\cdot 421^{3} + 130\cdot 421^{4} + 194\cdot 421^{5} + 405\cdot 421^{6} +O(421^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 138 + 164\cdot 421 + 368\cdot 421^{2} + 365\cdot 421^{3} + 85\cdot 421^{4} + 53\cdot 421^{5} + 184\cdot 421^{6} +O(421^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 189 + 279\cdot 421 + 121\cdot 421^{2} + 145\cdot 421^{3} + 154\cdot 421^{5} + 9\cdot 421^{6} +O(421^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 208 + 73\cdot 421 + 19\cdot 421^{2} + 285\cdot 421^{3} + 286\cdot 421^{4} + 123\cdot 421^{5} + 313\cdot 421^{6} +O(421^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 213 + 347\cdot 421 + 401\cdot 421^{2} + 135\cdot 421^{3} + 134\cdot 421^{4} + 297\cdot 421^{5} + 107\cdot 421^{6} +O(421^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 232 + 141\cdot 421 + 299\cdot 421^{2} + 275\cdot 421^{3} + 420\cdot 421^{4} + 266\cdot 421^{5} + 411\cdot 421^{6} +O(421^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 283 + 256\cdot 421 + 52\cdot 421^{2} + 55\cdot 421^{3} + 335\cdot 421^{4} + 367\cdot 421^{5} + 236\cdot 421^{6} +O(421^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 413 + 307\cdot 421 + 143\cdot 421^{2} + 290\cdot 421^{3} + 290\cdot 421^{4} + 226\cdot 421^{5} + 15\cdot 421^{6} +O(421^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6,8,3)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,4,7,5)(3,6)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,5,7,4)(3,6)$ | $\zeta_{4} + 1$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,4,6,7,8,5,3,2)$ | $0$ |
| $4$ | $8$ | $(1,7,3,4,8,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.