Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 + 156\cdot 211 + 83\cdot 211^{2} + 64\cdot 211^{3} + 129\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 + 160\cdot 211 + 70\cdot 211^{2} + 100\cdot 211^{3} + 147\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 81 + 52\cdot 211 + 204\cdot 211^{2} + 9\cdot 211^{3} + 68\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 108 + 115\cdot 211 + 156\cdot 211^{2} + 126\cdot 211^{3} + 112\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 163 + 93\cdot 211 + 201\cdot 211^{2} + 184\cdot 211^{3} + 93\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 186 + 184\cdot 211 + 134\cdot 211^{2} + 19\cdot 211^{3} + 205\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 199 + 200\cdot 211 + 110\cdot 211^{2} + 130\cdot 211^{3} + 32\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 204 + 90\cdot 211 + 92\cdot 211^{2} + 207\cdot 211^{3} + 54\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,6)(5,7)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,7,6)(2,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
