Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.4.233280000000.3 |
| 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 \(\Q(\sqrt{-3}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 120x^{4} - 360x^{2} + 720 \)
|
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 + 133\cdot 181 + 121\cdot 181^{2} + 88\cdot 181^{3} + 115\cdot 181^{4} + 42\cdot 181^{5} + 143\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 + 69\cdot 181 + 101\cdot 181^{2} + 17\cdot 181^{3} + 123\cdot 181^{4} + 45\cdot 181^{5} + 173\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 67 + 113\cdot 181 + 139\cdot 181^{2} + 57\cdot 181^{3} + 180\cdot 181^{4} + 125\cdot 181^{5} + 103\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 + 170\cdot 181 + 134\cdot 181^{2} + 58\cdot 181^{3} + 178\cdot 181^{4} + 76\cdot 181^{5} + 137\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 112 + 10\cdot 181 + 46\cdot 181^{2} + 122\cdot 181^{3} + 2\cdot 181^{4} + 104\cdot 181^{5} + 43\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 114 + 67\cdot 181 + 41\cdot 181^{2} + 123\cdot 181^{3} + 55\cdot 181^{5} + 77\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 119 + 111\cdot 181 + 79\cdot 181^{2} + 163\cdot 181^{3} + 57\cdot 181^{4} + 135\cdot 181^{5} + 7\cdot 181^{6} +O(181^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 149 + 47\cdot 181 + 59\cdot 181^{2} + 92\cdot 181^{3} + 65\cdot 181^{4} + 138\cdot 181^{5} + 37\cdot 181^{6} +O(181^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $0$ |
| $2$ | $8$ | $(1,6,7,5,8,3,2,4)$ | $0$ |
| $2$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $0$ |
| $2$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.