Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 91\cdot 109 + 93\cdot 109^{2} + 93\cdot 109^{3} + 84\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 69\cdot 109 + 39\cdot 109^{2} + 35\cdot 109^{3} + 7\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 92\cdot 109 + 83\cdot 109^{2} + 104\cdot 109^{3} + 10\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 38\cdot 109 + 79\cdot 109^{2} + 62\cdot 109^{3} + 66\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 70\cdot 109 + 29\cdot 109^{2} + 46\cdot 109^{3} + 42\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 16\cdot 109 + 25\cdot 109^{2} + 4\cdot 109^{3} + 98\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 39\cdot 109 + 69\cdot 109^{2} + 73\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 91 + 17\cdot 109 + 15\cdot 109^{2} + 15\cdot 109^{3} + 24\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,2,5,3)(4,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,5,3)(4,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
