Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 107\cdot 109 + 106\cdot 109^{2} + 35\cdot 109^{3} + 8\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 26\cdot 109 + 18\cdot 109^{2} + 100\cdot 109^{3} + 89\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 30\cdot 109 + 29\cdot 109^{2} + 47\cdot 109^{3} + 4\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 95\cdot 109 + 98\cdot 109^{2} + 93\cdot 109^{3} + 54\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 13\cdot 109 + 10\cdot 109^{2} + 15\cdot 109^{3} + 54\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 70 + 78\cdot 109 + 79\cdot 109^{2} + 61\cdot 109^{3} + 104\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 82\cdot 109 + 90\cdot 109^{2} + 8\cdot 109^{3} + 19\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 102 + 109 + 2\cdot 109^{2} + 73\cdot 109^{3} + 100\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,2,6)(3,8,5,7)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
