Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 50\cdot 53 + 43\cdot 53^{2} + 35\cdot 53^{3} + 4\cdot 53^{4} + 31\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 40\cdot 53 + 40\cdot 53^{2} + 6\cdot 53^{3} + 44\cdot 53^{4} + 23\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 11\cdot 53 + 39\cdot 53^{2} + 32\cdot 53^{3} + 8\cdot 53^{4} + 19\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 35\cdot 53 + 35\cdot 53^{2} + 5\cdot 53^{3} + 43\cdot 53^{4} + 47\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 17\cdot 53 + 17\cdot 53^{2} + 47\cdot 53^{3} + 9\cdot 53^{4} + 5\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 41\cdot 53 + 13\cdot 53^{2} + 20\cdot 53^{3} + 44\cdot 53^{4} + 33\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 + 12\cdot 53 + 12\cdot 53^{2} + 46\cdot 53^{3} + 8\cdot 53^{4} + 29\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 49 + 2\cdot 53 + 9\cdot 53^{2} + 17\cdot 53^{3} + 48\cdot 53^{4} + 21\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,6,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
