Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 117 + 82\cdot 347 + 296\cdot 347^{2} + 250\cdot 347^{3} + 205\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 140 + 167\cdot 347 + 65\cdot 347^{2} + 73\cdot 347^{3} + 214\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 196 + 318\cdot 347 + 93\cdot 347^{2} + 153\cdot 347^{3} + 297\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 241 + 125\cdot 347 + 238\cdot 347^{2} + 216\cdot 347^{3} + 323\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.
