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 8.4.233280000000.5 |
| 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 $\Q_{ 151 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 85\cdot 151^{2} + 76\cdot 151^{3} + 139\cdot 151^{4} + 6\cdot 151^{5} + 43\cdot 151^{6} + 145\cdot 151^{7} +O(151^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 42\cdot 151 + 77\cdot 151^{2} + 68\cdot 151^{3} + 106\cdot 151^{4} + 132\cdot 151^{5} + 140\cdot 151^{6} + 121\cdot 151^{7} +O(151^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 26\cdot 151 + 132\cdot 151^{2} + 28\cdot 151^{3} + 91\cdot 151^{4} + 108\cdot 151^{5} + 13\cdot 151^{6} + 53\cdot 151^{7} +O(151^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 65 + 151 + 21\cdot 151^{2} + 21\cdot 151^{3} + 47\cdot 151^{4} + 88\cdot 151^{5} + 144\cdot 151^{6} + 110\cdot 151^{7} +O(151^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 86 + 149\cdot 151 + 129\cdot 151^{2} + 129\cdot 151^{3} + 103\cdot 151^{4} + 62\cdot 151^{5} + 6\cdot 151^{6} + 40\cdot 151^{7} +O(151^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 129 + 124\cdot 151 + 18\cdot 151^{2} + 122\cdot 151^{3} + 59\cdot 151^{4} + 42\cdot 151^{5} + 137\cdot 151^{6} + 97\cdot 151^{7} +O(151^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 136 + 108\cdot 151 + 73\cdot 151^{2} + 82\cdot 151^{3} + 44\cdot 151^{4} + 18\cdot 151^{5} + 10\cdot 151^{6} + 29\cdot 151^{7} +O(151^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 144 + 150\cdot 151 + 65\cdot 151^{2} + 74\cdot 151^{3} + 11\cdot 151^{4} + 144\cdot 151^{5} + 107\cdot 151^{6} + 5\cdot 151^{7} +O(151^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,7,6,8,4,2,3)$ | $0$ |
| $2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.