Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 48\cdot 53 + 9\cdot 53^{2} + 48\cdot 53^{3} + 49\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 30\cdot 53 + 50\cdot 53^{2} + 12\cdot 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 32\cdot 53 + 41\cdot 53^{2} + 37\cdot 53^{3} + 45\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 16\cdot 53 + 24\cdot 53^{2} + 25\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 36\cdot 53 + 28\cdot 53^{2} + 27\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 20\cdot 53 + 11\cdot 53^{2} + 15\cdot 53^{3} + 7\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 42 + 22\cdot 53 + 2\cdot 53^{2} + 40\cdot 53^{3} + 51\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 4\cdot 53 + 43\cdot 53^{2} + 4\cdot 53^{3} + 3\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,5)(2,3,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
