Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 12\cdot 109 + 75\cdot 109^{2} + 81\cdot 109^{3} + 86\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 89\cdot 109 + 51\cdot 109^{2} + 53\cdot 109^{3} + 57\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 62\cdot 109 + 77\cdot 109^{2} + 45\cdot 109^{3} + 53\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 17\cdot 109 + 10\cdot 109^{2} + 100\cdot 109^{3} + 106\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 63 + 21\cdot 109 + 20\cdot 109^{2} + 31\cdot 109^{3} + 37\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 92 + 90\cdot 109 + 103\cdot 109^{2} + 82\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 + 20\cdot 109 + 66\cdot 109^{2} + 94\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 102 + 12\cdot 109 + 31\cdot 109^{2} + 40\cdot 109^{3} + 7\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,6,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,8,2,7)(3,5,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$-2$ |
| $2$ |
$4$ |
$(1,3,2,4)(5,8,6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,2,7)(3,5,4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,2,6)(3,7,4,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
