Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 152\cdot 197 + 50\cdot 197^{2} + 185\cdot 197^{3} + 177\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 88\cdot 197 + 195\cdot 197^{2} + 33\cdot 197^{3} + 153\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 + 163\cdot 197 + 131\cdot 197^{2} + 96\cdot 197^{3} + 86\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 + 159\cdot 197 + 63\cdot 197^{2} + 154\cdot 197^{3} + 114\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 122 + 26\cdot 197 + 109\cdot 197^{2} + 120\cdot 197^{3} + 183\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 128 + 101\cdot 197 + 45\cdot 197^{2} + 183\cdot 197^{3} + 116\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 132 + 44\cdot 197 + 89\cdot 197^{2} + 22\cdot 197^{3} + 9\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 193 + 51\cdot 197 + 102\cdot 197^{2} + 188\cdot 197^{3} + 142\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,3,8,5)(2,7,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,6)(3,5)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,3)(4,8)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,5)(2,7,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
