Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 + 319\cdot 467 + 166\cdot 467^{2} + 90\cdot 467^{3} + 336\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 284 + 361\cdot 467 + 371\cdot 467^{2} + 102\cdot 467^{3} + 444\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 304 + 320\cdot 467 + 86\cdot 467^{2} + 383\cdot 467^{3} + 459\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 371 + 34\cdot 467 + 453\cdot 467^{2} + 58\cdot 467^{3} + 426\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 387 + 364\cdot 467 + 322\cdot 467^{2} + 298\cdot 467^{3} + 201\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
