Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(6400\)\(\medspace = 2^{8} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.4.327680000000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 60x^{4} + 160x^{2} + 20 \)
|
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 + 106\cdot 131 + 70\cdot 131^{2} + 55\cdot 131^{3} + 102\cdot 131^{4} + 6\cdot 131^{5} + 96\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 + 14\cdot 131 + 32\cdot 131^{2} + 83\cdot 131^{3} + 97\cdot 131^{4} + 87\cdot 131^{5} + 51\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 + 3\cdot 131 + 108\cdot 131^{2} + 3\cdot 131^{3} + 38\cdot 131^{4} + 22\cdot 131^{5} + 4\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 + 45\cdot 131 + 57\cdot 131^{2} + 44\cdot 131^{3} + 4\cdot 131^{4} + 53\cdot 131^{5} + 71\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 68 + 85\cdot 131 + 73\cdot 131^{2} + 86\cdot 131^{3} + 126\cdot 131^{4} + 77\cdot 131^{5} + 59\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 70 + 127\cdot 131 + 22\cdot 131^{2} + 127\cdot 131^{3} + 92\cdot 131^{4} + 108\cdot 131^{5} + 126\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 98 + 116\cdot 131 + 98\cdot 131^{2} + 47\cdot 131^{3} + 33\cdot 131^{4} + 43\cdot 131^{5} + 79\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 103 + 24\cdot 131 + 60\cdot 131^{2} + 75\cdot 131^{3} + 28\cdot 131^{4} + 124\cdot 131^{5} + 34\cdot 131^{6} +O(131^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
| $2$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $0$ |
| $2$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.