Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 89\cdot 109 + 73\cdot 109^{2} + 18\cdot 109^{3} + 20\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 50\cdot 109 + 78\cdot 109^{2} + 53\cdot 109^{3} + 97\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 59\cdot 109 + 53\cdot 109^{2} + 52\cdot 109^{3} + 86\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 28\cdot 109 + 46\cdot 109^{2} + 3\cdot 109^{3} + 75\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 62 + 80\cdot 109 + 62\cdot 109^{2} + 105\cdot 109^{3} + 33\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 + 49\cdot 109 + 55\cdot 109^{2} + 56\cdot 109^{3} + 22\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 84 + 58\cdot 109 + 30\cdot 109^{2} + 55\cdot 109^{3} + 11\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 100 + 19\cdot 109 + 35\cdot 109^{2} + 90\cdot 109^{3} + 88\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,6)(3,8,4,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,8,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
