Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 79\cdot 109 + 91\cdot 109^{2} + 44\cdot 109^{3} + 4\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 99\cdot 109 + 34\cdot 109^{2} + 71\cdot 109^{3} + 36\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 + 47\cdot 109 + 22\cdot 109^{2} + 46\cdot 109^{3} + 90\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 75\cdot 109 + 69\cdot 109^{2} + 88\cdot 109^{3} + 73\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 15\cdot 109 + 34\cdot 109^{2} + 38\cdot 109^{3} + 49\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 92 + 23\cdot 109 + 57\cdot 109^{2} + 63\cdot 109^{3} + 18\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 + 72\cdot 109 + 21\cdot 109^{2} + 13\cdot 109^{3} + 103\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 107 + 21\cdot 109 + 104\cdot 109^{2} + 69\cdot 109^{3} + 59\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,7,4)(3,5,6,8)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,6)(3,4)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,2,7,4)(3,5,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
