Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 + 167\cdot 467 + 58\cdot 467^{2} + 389\cdot 467^{3} + 407\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 272 + 204\cdot 467 + 155\cdot 467^{2} + 14\cdot 467^{3} + 376\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 308 + 20\cdot 467 + 320\cdot 467^{2} + 154\cdot 467^{3} + 164\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 318 + 174\cdot 467 + 217\cdot 467^{2} + 306\cdot 467^{3} + 317\cdot 467^{4} +O\left(467^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 446 + 366\cdot 467 + 182\cdot 467^{2} + 69\cdot 467^{3} + 135\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
