Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 155\cdot 211 + 23\cdot 211^{2} + 30\cdot 211^{3} + 195\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 161\cdot 211 + 50\cdot 211^{2} + 194\cdot 211^{3} + 58\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 + 112\cdot 211 + 167\cdot 211^{2} + 60\cdot 211^{3} + 148\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 104 + 204\cdot 211 + 179\cdot 211^{2} + 136\cdot 211^{3} + 19\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
