Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 21\cdot 107 + 84\cdot 107^{2} + 57\cdot 107^{3} + 91\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 63\cdot 107 + 101\cdot 107^{2} + 54\cdot 107^{3} + 90\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 37\cdot 107 + 87\cdot 107^{2} + 46\cdot 107^{3} + 100\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 102 + 91\cdot 107 + 47\cdot 107^{2} + 54\cdot 107^{3} + 38\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $4$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
