Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 185\cdot 227 + 20\cdot 227^{2} + 128\cdot 227^{3} + 116\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 61 + 99\cdot 227 + 94\cdot 227^{2} + 182\cdot 227^{3} + 44\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 + 159\cdot 227 + 196\cdot 227^{2} + 96\cdot 227^{3} + 51\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 70 + 10\cdot 227 + 142\cdot 227^{2} + 46\cdot 227^{3} + 14\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 157 + 216\cdot 227 + 84\cdot 227^{2} + 180\cdot 227^{3} + 212\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 161 + 67\cdot 227 + 30\cdot 227^{2} + 130\cdot 227^{3} + 175\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 166 + 127\cdot 227 + 132\cdot 227^{2} + 44\cdot 227^{3} + 182\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 197 + 41\cdot 227 + 206\cdot 227^{2} + 98\cdot 227^{3} + 110\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,3)(5,6,8,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
