Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 353 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 325\cdot 353 + 227\cdot 353^{2} + 117\cdot 353^{3} + 245\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 186\cdot 353 + 86\cdot 353^{2} + 57\cdot 353^{3} + 265\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 123 + 311\cdot 353 + 197\cdot 353^{2} + 2\cdot 353^{3} + 189\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 205 + 236\cdot 353 + 193\cdot 353^{2} + 175\cdot 353^{3} + 6\cdot 353^{4} +O\left(353^{ 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.
