Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 77\cdot 109 + 84\cdot 109^{2} + 93\cdot 109^{3} + 41\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 68\cdot 109 + 86\cdot 109^{2} + 89\cdot 109^{3} + 83\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 15\cdot 109 + 88\cdot 109^{2} + 12\cdot 109^{3} + 89\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 81\cdot 109 + 73\cdot 109^{2} + 100\cdot 109^{3} + 100\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 + 27\cdot 109 + 35\cdot 109^{2} + 8\cdot 109^{3} + 8\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 93\cdot 109 + 20\cdot 109^{2} + 96\cdot 109^{3} + 19\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 90 + 40\cdot 109 + 22\cdot 109^{2} + 19\cdot 109^{3} + 25\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 95 + 31\cdot 109 + 24\cdot 109^{2} + 15\cdot 109^{3} + 67\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,5,6)(3,8,7,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,5,6)(3,8,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
