Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 21\cdot 53 + 4\cdot 53^{2} + 19\cdot 53^{3} + 36\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 26\cdot 53 + 16\cdot 53^{2} + 38\cdot 53^{3} + 7\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 2\cdot 53 + 41\cdot 53^{2} + 33\cdot 53^{3} + 38\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 3\cdot 53 + 44\cdot 53^{2} + 14\cdot 53^{3} + 23\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 39\cdot 53 + 20\cdot 53^{2} + 5\cdot 53^{3} + 44\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 44\cdot 53 + 32\cdot 53^{2} + 24\cdot 53^{3} + 15\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 + 36\cdot 53 + 24\cdot 53^{2} + 47\cdot 53^{3} + 30\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 50 + 37\cdot 53 + 27\cdot 53^{2} + 28\cdot 53^{3} + 15\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,6)(3,8)(4,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,2,4)(5,7,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,8)(4,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,4)(5,7,6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
