Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 46\cdot 53 + 29\cdot 53^{2} + 27\cdot 53^{3} + 47\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 51\cdot 53 + 27\cdot 53^{2} + 11\cdot 53^{3} + 48\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 50\cdot 53 + 14\cdot 53^{2} + 32\cdot 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 19\cdot 53 + 44\cdot 53^{3} + 51\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 + 15\cdot 53 + 14\cdot 53^{2} + 52\cdot 53^{3} + 40\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 42\cdot 53 + 7\cdot 53^{2} + 2\cdot 53^{3} + 5\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 46 + 10\cdot 53 + 33\cdot 53^{2} + 34\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 51 + 27\cdot 53 + 30\cdot 53^{2} + 7\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,6)(3,7,5,4)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,6)(3,7,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
