Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.2097152000.5 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.40.4t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of \(\Q(\sqrt[4]{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{4} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 127\cdot 269 + 243\cdot 269^{2} + 31\cdot 269^{3} + 54\cdot 269^{4} + 269^{5} +O(269^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 + 150\cdot 269 + 255\cdot 269^{2} + 209\cdot 269^{3} + 187\cdot 269^{4} + 166\cdot 269^{5} +O(269^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 + 202\cdot 269 + 77\cdot 269^{2} + 38\cdot 269^{3} + 188\cdot 269^{4} + 157\cdot 269^{5} +O(269^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 134 + 268\cdot 269 + 146\cdot 269^{2} + 40\cdot 269^{3} + 79\cdot 269^{4} + 27\cdot 269^{5} +O(269^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 135 + 122\cdot 269^{2} + 228\cdot 269^{3} + 189\cdot 269^{4} + 241\cdot 269^{5} +O(269^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 226 + 66\cdot 269 + 191\cdot 269^{2} + 230\cdot 269^{3} + 80\cdot 269^{4} + 111\cdot 269^{5} +O(269^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 228 + 118\cdot 269 + 13\cdot 269^{2} + 59\cdot 269^{3} + 81\cdot 269^{4} + 102\cdot 269^{5} +O(269^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 240 + 141\cdot 269 + 25\cdot 269^{2} + 237\cdot 269^{3} + 214\cdot 269^{4} + 267\cdot 269^{5} +O(269^{6})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ | |
| $4$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ | ✓ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(2,5,7,4)$ | $\zeta_{4} + 1$ | |
| $2$ | $4$ | $(2,4,7,5)$ | $-\zeta_{4} + 1$ | |
| $2$ | $4$ | $(1,8)(2,4,7,5)(3,6)$ | $-\zeta_{4} - 1$ | |
| $2$ | $4$ | $(1,8)(2,5,7,4)(3,6)$ | $\zeta_{4} - 1$ | |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ | |
| $4$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ | |
| $4$ | $8$ | $(1,2,3,4,8,7,6,5)$ | $0$ | |
| $4$ | $8$ | $(1,4,6,2,8,5,3,7)$ | $0$ |