Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 13.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 43\cdot 53 + 21\cdot 53^{2} + 6\cdot 53^{3} + 21\cdot 53^{4} + 46\cdot 53^{5} + 40\cdot 53^{6} + 4\cdot 53^{7} + 52\cdot 53^{8} + 50\cdot 53^{9} + 44\cdot 53^{10} + 53^{11} + 16\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 44\cdot 53 + 31\cdot 53^{2} + 37\cdot 53^{3} + 30\cdot 53^{4} + 5\cdot 53^{5} + 14\cdot 53^{6} + 47\cdot 53^{7} + 42\cdot 53^{8} + 8\cdot 53^{9} + 33\cdot 53^{10} + 9\cdot 53^{11} + 29\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 7\cdot 53 + 34\cdot 53^{2} + 40\cdot 53^{3} + 15\cdot 53^{4} + 24\cdot 53^{6} + 4\cdot 53^{7} + 45\cdot 53^{8} + 33\cdot 53^{9} + 5\cdot 53^{10} + 9\cdot 53^{11} + 41\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 34\cdot 53 + 45\cdot 53^{2} + 31\cdot 53^{3} + 10\cdot 53^{4} + 6\cdot 53^{5} + 18\cdot 53^{6} + 34\cdot 53^{7} + 32\cdot 53^{8} + 11\cdot 53^{9} + 40\cdot 53^{10} + 6\cdot 53^{11} + 5\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 18\cdot 53 + 7\cdot 53^{2} + 21\cdot 53^{3} + 42\cdot 53^{4} + 46\cdot 53^{5} + 34\cdot 53^{6} + 18\cdot 53^{7} + 20\cdot 53^{8} + 41\cdot 53^{9} + 12\cdot 53^{10} + 46\cdot 53^{11} + 47\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 45\cdot 53 + 18\cdot 53^{2} + 12\cdot 53^{3} + 37\cdot 53^{4} + 52\cdot 53^{5} + 28\cdot 53^{6} + 48\cdot 53^{7} + 7\cdot 53^{8} + 19\cdot 53^{9} + 47\cdot 53^{10} + 43\cdot 53^{11} + 11\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 8\cdot 53 + 21\cdot 53^{2} + 15\cdot 53^{3} + 22\cdot 53^{4} + 47\cdot 53^{5} + 38\cdot 53^{6} + 5\cdot 53^{7} + 10\cdot 53^{8} + 44\cdot 53^{9} + 19\cdot 53^{10} + 43\cdot 53^{11} + 23\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 9\cdot 53 + 31\cdot 53^{2} + 46\cdot 53^{3} + 31\cdot 53^{4} + 6\cdot 53^{5} + 12\cdot 53^{6} + 48\cdot 53^{7} + 2\cdot 53^{9} + 8\cdot 53^{10} + 51\cdot 53^{11} + 36\cdot 53^{12} +O\left(53^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2,8,7)(3,5,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
