Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 419 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 + 292\cdot 419 + 258\cdot 419^{2} + 43\cdot 419^{3} + 392\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 154 + 267\cdot 419 + 382\cdot 419^{2} + 205\cdot 419^{3} + 281\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 319 + 349\cdot 419 + 380\cdot 419^{2} + 53\cdot 419^{3} + 319\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 321 + 347\cdot 419 + 234\cdot 419^{2} + 115\cdot 419^{3} + 264\cdot 419^{4} +O\left(419^{ 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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $6$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
