Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 35\cdot 197 + 165\cdot 197^{2} + 58\cdot 197^{3} + 14\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 85 + 40\cdot 197 + 153\cdot 197^{2} + 25\cdot 197^{3} + 182\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 154 + 93\cdot 197 + 186\cdot 197^{2} + 155\cdot 197^{3} + 60\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 164 + 150\cdot 197 + 104\cdot 197^{2} + 75\cdot 197^{3} + 110\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 187 + 73\cdot 197 + 178\cdot 197^{2} + 77\cdot 197^{3} + 26\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
