Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 263 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 203\cdot 263 + 22\cdot 263^{2} + 129\cdot 263^{3} + 229\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 + 137\cdot 263 + 38\cdot 263^{2} + 116\cdot 263^{3} + 57\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 80 + 215\cdot 263 + 78\cdot 263^{2} + 82\cdot 263^{3} + 224\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 99 + 211\cdot 263 + 143\cdot 263^{2} + 253\cdot 263^{3} + 128\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 238 + 21\cdot 263 + 242\cdot 263^{2} + 207\cdot 263^{3} + 148\cdot 263^{4} +O\left(263^{ 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.
