Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 + 20\cdot 347 + 298\cdot 347^{2} + 271\cdot 347^{3} + 23\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 147 + 126\cdot 347 + 34\cdot 347^{2} + 146\cdot 347^{3} + 245\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 204 + 108\cdot 347 + 151\cdot 347^{2} + 270\cdot 347^{3} + 263\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 311 + 91\cdot 347 + 210\cdot 347^{2} + 5\cdot 347^{3} + 161\cdot 347^{4} +O\left(347^{ 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.
