Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.4.5120000000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.40.4t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 15x^{4} + 10x^{2} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 24\cdot 59 + 25\cdot 59^{2} + 3\cdot 59^{3} + 46\cdot 59^{4} + 18\cdot 59^{5} + 14\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 17\cdot 59 + 19\cdot 59^{2} + 6\cdot 59^{3} + 46\cdot 59^{4} + 10\cdot 59^{5} + 54\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 45\cdot 59 + 21\cdot 59^{2} + 59^{3} + 41\cdot 59^{4} + 4\cdot 59^{5} + 40\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 31\cdot 59 + 55\cdot 59^{2} + 56\cdot 59^{3} + 5\cdot 59^{4} + 44\cdot 59^{5} + 23\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 + 27\cdot 59 + 3\cdot 59^{2} + 2\cdot 59^{3} + 53\cdot 59^{4} + 14\cdot 59^{5} + 35\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 + 13\cdot 59 + 37\cdot 59^{2} + 57\cdot 59^{3} + 17\cdot 59^{4} + 54\cdot 59^{5} + 18\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 41\cdot 59 + 39\cdot 59^{2} + 52\cdot 59^{3} + 12\cdot 59^{4} + 48\cdot 59^{5} + 4\cdot 59^{6} +O(59^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 41 + 34\cdot 59 + 33\cdot 59^{2} + 55\cdot 59^{3} + 12\cdot 59^{4} + 40\cdot 59^{5} + 44\cdot 59^{6} +O(59^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $0$ |
| $2$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
| $2$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.