Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 373 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 55 + 118\cdot 373 + 172\cdot 373^{2} + 133\cdot 373^{3} + 203\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 152 + 357\cdot 373 + 30\cdot 373^{2} + 337\cdot 373^{3} + 97\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 259 + 274\cdot 373 + 241\cdot 373^{2} + 101\cdot 373^{3} + 19\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 280 + 368\cdot 373 + 300\cdot 373^{2} + 173\cdot 373^{3} + 52\cdot 373^{4} +O\left(373^{ 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.
