Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 64\cdot 109 + 96\cdot 109^{2} + 14\cdot 109^{3} + 30\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 66\cdot 109 + 102\cdot 109^{2} + 77\cdot 109^{3} + 90\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 88\cdot 109 + 75\cdot 109^{2} + 99\cdot 109^{3} + 80\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 33\cdot 109 + 94\cdot 109^{2} + 78\cdot 109^{3} + 87\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 75\cdot 109 + 14\cdot 109^{2} + 30\cdot 109^{3} + 21\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 + 20\cdot 109 + 33\cdot 109^{2} + 9\cdot 109^{3} + 28\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 42\cdot 109 + 6\cdot 109^{2} + 31\cdot 109^{3} + 18\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 89 + 44\cdot 109 + 12\cdot 109^{2} + 94\cdot 109^{3} + 78\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,5)(3,4,8,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,6,5)(3,4,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
