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 8.4.414720000000.1 |
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
$f(x)$ | $=$ | \( x^{8} - 135x^{4} - 270x^{2} + 405 \) . |
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 8.
Roots:
$r_{ 1 }$ | $=$ |
\( 29 + 120\cdot 139 + 138\cdot 139^{2} + 55\cdot 139^{3} + 7\cdot 139^{4} + 42\cdot 139^{5} + 84\cdot 139^{6} + 3\cdot 139^{7} +O(139^{8})\)
$r_{ 2 }$ |
$=$ |
\( 38 + 106\cdot 139 + 17\cdot 139^{2} + 44\cdot 139^{3} + 55\cdot 139^{4} + 42\cdot 139^{5} + 38\cdot 139^{6} + 58\cdot 139^{7} +O(139^{8})\)
| $r_{ 3 }$ |
$=$ |
\( 41 + 7\cdot 139 + 101\cdot 139^{2} + 137\cdot 139^{3} + 136\cdot 139^{4} + 80\cdot 139^{5} + 33\cdot 139^{6} + 62\cdot 139^{7} +O(139^{8})\)
| $r_{ 4 }$ |
$=$ |
\( 64 + 121\cdot 139 + 15\cdot 139^{2} + 18\cdot 139^{3} + 60\cdot 139^{4} + 13\cdot 139^{5} + 40\cdot 139^{6} + 28\cdot 139^{7} +O(139^{8})\)
| $r_{ 5 }$ |
$=$ |
\( 75 + 17\cdot 139 + 123\cdot 139^{2} + 120\cdot 139^{3} + 78\cdot 139^{4} + 125\cdot 139^{5} + 98\cdot 139^{6} + 110\cdot 139^{7} +O(139^{8})\)
| $r_{ 6 }$ |
$=$ |
\( 98 + 131\cdot 139 + 37\cdot 139^{2} + 139^{3} + 2\cdot 139^{4} + 58\cdot 139^{5} + 105\cdot 139^{6} + 76\cdot 139^{7} +O(139^{8})\)
| $r_{ 7 }$ |
$=$ |
\( 101 + 32\cdot 139 + 121\cdot 139^{2} + 94\cdot 139^{3} + 83\cdot 139^{4} + 96\cdot 139^{5} + 100\cdot 139^{6} + 80\cdot 139^{7} +O(139^{8})\)
| $r_{ 8 }$ |
$=$ |
\( 110 + 18\cdot 139 + 83\cdot 139^{3} + 131\cdot 139^{4} + 96\cdot 139^{5} + 54\cdot 139^{6} + 135\cdot 139^{7} +O(139^{8})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,8)(4,5)$ | $0$ |
$1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
$2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $0$ |
$2$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $0$ |
$2$ | $8$ | $(1,6,4,7,8,3,5,2)$ | $0$ |
$2$ | $8$ | $(1,7,5,6,8,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.