Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 + 116\cdot 223 + 74\cdot 223^{2} + 150\cdot 223^{3} + 169\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 + 111\cdot 223 + 101\cdot 223^{2} + 223^{3} + 89\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 96 + 112\cdot 223 + 157\cdot 223^{2} + 128\cdot 223^{3} + 70\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 110 + 201\cdot 223 + 112\cdot 223^{2} + 59\cdot 223^{3} + 103\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 113 + 21\cdot 223 + 110\cdot 223^{2} + 163\cdot 223^{3} + 119\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 127 + 110\cdot 223 + 65\cdot 223^{2} + 94\cdot 223^{3} + 152\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 170 + 111\cdot 223 + 121\cdot 223^{2} + 221\cdot 223^{3} + 133\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 174 + 106\cdot 223 + 148\cdot 223^{2} + 72\cdot 223^{3} + 53\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
