Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 17\cdot 101 + 12\cdot 101^{2} + 26\cdot 101^{3} + 87\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 48\cdot 101 + 11\cdot 101^{2} + 95\cdot 101^{3} + 33\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 68\cdot 101 + 80\cdot 101^{2} + 13\cdot 101^{3} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 14\cdot 101 + 25\cdot 101^{2} + 57\cdot 101^{3} + 85\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 91 + 53\cdot 101 + 72\cdot 101^{2} + 9\cdot 101^{3} + 96\cdot 101^{4} +O\left(101^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
