Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 85\cdot 109 + 89\cdot 109^{2} + 107\cdot 109^{3} + 24\cdot 109^{4} + 88\cdot 109^{5} +O\left(109^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 46\cdot 109 + 93\cdot 109^{2} + 61\cdot 109^{3} + 59\cdot 109^{4} + 72\cdot 109^{5} +O\left(109^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 108\cdot 109 + 81\cdot 109^{2} + 82\cdot 109^{3} + 99\cdot 109^{4} + 12\cdot 109^{5} +O\left(109^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 + 41\cdot 109 + 103\cdot 109^{2} + 43\cdot 109^{3} + 108\cdot 109^{4} + 108\cdot 109^{5} +O\left(109^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 67\cdot 109 + 5\cdot 109^{2} + 65\cdot 109^{3} +O\left(109^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 85 + 27\cdot 109^{2} + 26\cdot 109^{3} + 9\cdot 109^{4} + 96\cdot 109^{5} +O\left(109^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 + 62\cdot 109 + 15\cdot 109^{2} + 47\cdot 109^{3} + 49\cdot 109^{4} + 36\cdot 109^{5} +O\left(109^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 103 + 23\cdot 109 + 19\cdot 109^{2} + 109^{3} + 84\cdot 109^{4} + 20\cdot 109^{5} +O\left(109^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
