Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 52\cdot 53 + 45\cdot 53^{2} + 4\cdot 53^{3} + 49\cdot 53^{4} + 50\cdot 53^{5} + 53^{6} + 33\cdot 53^{7} + 29\cdot 53^{8} + 4\cdot 53^{9} + 48\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 17\cdot 53 + 19\cdot 53^{2} + 47\cdot 53^{3} + 3\cdot 53^{4} + 8\cdot 53^{5} + 2\cdot 53^{6} + 38\cdot 53^{7} + 48\cdot 53^{8} + 20\cdot 53^{9} + 36\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 4\cdot 53 + 31\cdot 53^{2} + 2\cdot 53^{3} + 19\cdot 53^{4} + 29\cdot 53^{5} + 24\cdot 53^{6} + 12\cdot 53^{7} + 30\cdot 53^{8} + 18\cdot 53^{9} + 48\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 29\cdot 53 + 44\cdot 53^{2} + 44\cdot 53^{3} + 45\cdot 53^{4} + 47\cdot 53^{5} + 9\cdot 53^{6} + 47\cdot 53^{7} + 14\cdot 53^{8} + 4\cdot 53^{9} + 28\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 23\cdot 53 + 8\cdot 53^{2} + 8\cdot 53^{3} + 7\cdot 53^{4} + 5\cdot 53^{5} + 43\cdot 53^{6} + 5\cdot 53^{7} + 38\cdot 53^{8} + 48\cdot 53^{9} + 24\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 + 48\cdot 53 + 21\cdot 53^{2} + 50\cdot 53^{3} + 33\cdot 53^{4} + 23\cdot 53^{5} + 28\cdot 53^{6} + 40\cdot 53^{7} + 22\cdot 53^{8} + 34\cdot 53^{9} + 4\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 35\cdot 53 + 33\cdot 53^{2} + 5\cdot 53^{3} + 49\cdot 53^{4} + 44\cdot 53^{5} + 50\cdot 53^{6} + 14\cdot 53^{7} + 4\cdot 53^{8} + 32\cdot 53^{9} + 16\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 51 + 7\cdot 53^{2} + 48\cdot 53^{3} + 3\cdot 53^{4} + 2\cdot 53^{5} + 51\cdot 53^{6} + 19\cdot 53^{7} + 23\cdot 53^{8} + 48\cdot 53^{9} + 4\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,3,7,8,5,6,2)$ |
| $(3,6)(4,5)$ |
| $(2,7)(4,5)$ |
| $(1,8)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $0$ |
| $4$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $0$ |
| $4$ | $8$ | $(1,4,6,7,8,5,3,2)$ | $0$ |
| $4$ | $8$ | $(1,7,3,4,8,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
