Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 49\cdot 53 + 42\cdot 53^{2} + 4\cdot 53^{3} + 15\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 51\cdot 53 + 15\cdot 53^{2} + 12\cdot 53^{3} + 27\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 12\cdot 53 + 34\cdot 53^{3} + 9\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 45\cdot 53 + 46\cdot 53^{2} + 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $4$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
