Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 53 + 39\cdot 53^{2} + 11\cdot 53^{3} + 5\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 35\cdot 53 + 45\cdot 53^{2} + 25\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 17\cdot 53 + 9\cdot 53^{2} + 7\cdot 53^{3} + 5\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 8\cdot 53 + 4\cdot 53^{2} + 37\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 26\cdot 53 + 50\cdot 53^{3} + 44\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 8\cdot 53 + 17\cdot 53^{2} + 31\cdot 53^{3} + 41\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 + 32\cdot 53 + 14\cdot 53^{2} + 39\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 + 29\cdot 53 + 28\cdot 53^{2} + 9\cdot 53^{3} + 42\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,6)(3,8,5,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,6)(3,5)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,5)(3,6)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,6)(3,8,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
