Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 99\cdot 107 + 51\cdot 107^{2} + 37\cdot 107^{3} + 80\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 + 21\cdot 107 + 25\cdot 107^{2} + 97\cdot 107^{3} + 45\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 65 + 28\cdot 107 + 78\cdot 107^{2} + 72\cdot 107^{3} + 71\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 92 + 64\cdot 107 + 58\cdot 107^{2} + 6\cdot 107^{3} + 16\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 value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
