Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 106\cdot 199 + 78\cdot 199^{2} + 61\cdot 199^{3} + 63\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 25\cdot 199 + 148\cdot 199^{2} + 111\cdot 199^{3} + 58\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 179\cdot 199 + 8\cdot 199^{2} + 14\cdot 199^{3} + 187\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 41\cdot 199 + 63\cdot 199^{2} + 50\cdot 199^{3} + 23\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 79 + 131\cdot 199 + 171\cdot 199^{2} + 154\cdot 199^{3} + 177\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 118 + 37\cdot 199 + 58\cdot 199^{2} + 80\cdot 199^{3} + 74\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 157 + 72\cdot 199 + 156\cdot 199^{2} + 13\cdot 199^{3} + 121\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 195 + 3\cdot 199 + 111\cdot 199^{2} + 110\cdot 199^{3} + 90\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,3,4)(2,5,6,7)$ |
| $(1,3)(2,6)(4,8)(5,7)$ |
| $(1,6,3,2)(4,7,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,6)(4,8)(5,7)$ |
$-2$ |
| $2$ |
$4$ |
$(1,6,3,2)(4,7,8,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,3,4)(2,5,6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,3,5)(2,8,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
