Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 353 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 113 + 146\cdot 353 + 304\cdot 353^{2} + 348\cdot 353^{3} + 275\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 152 + 112\cdot 353 + 231\cdot 353^{2} + 243\cdot 353^{3} + 275\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 164 + 221\cdot 353 + 253\cdot 353^{2} + 199\cdot 353^{3} + 171\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 292 + 197\cdot 353 + 331\cdot 353^{2} + 273\cdot 353^{3} + 3\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 338 + 27\cdot 353 + 291\cdot 353^{2} + 345\cdot 353^{3} + 331\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
