Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 9\cdot 53 + 14\cdot 53^{2} + 13\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 22\cdot 53 + 17\cdot 53^{2} + 32\cdot 53^{3} + 44\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 42\cdot 53 + 30\cdot 53^{2} + 32\cdot 53^{3} + 42\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 38\cdot 53 + 28\cdot 53^{2} + 14\cdot 53^{3} + 20\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 35\cdot 53 + 27\cdot 53^{2} + 13\cdot 53^{3} + 27\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 38\cdot 53 + 46\cdot 53^{2} + 46\cdot 53^{3} + 36\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 43 + 38\cdot 53 + 32\cdot 53^{2} + 18\cdot 53^{3} + 25\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 + 38\cdot 53 + 13\cdot 53^{2} + 40\cdot 53^{3} + 17\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,5,6)(3,4,8,7)$ |
| $(1,3)(2,7)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,6)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,6)(3,4,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
