Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 19\cdot 53 + 34\cdot 53^{2} + 35\cdot 53^{3} + 2\cdot 53^{4} + 48\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 21\cdot 53 + 12\cdot 53^{2} + 23\cdot 53^{3} + 31\cdot 53^{4} + 25\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 43\cdot 53 + 8\cdot 53^{2} + 13\cdot 53^{3} + 17\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 22\cdot 53 + 50\cdot 53^{2} + 33\cdot 53^{3} + 18\cdot 53^{4} + 15\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 31\cdot 53 + 45\cdot 53^{2} + 47\cdot 53^{3} + 9\cdot 53^{4} + 10\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 35\cdot 53 + 3\cdot 53^{2} + 50\cdot 53^{3} + 36\cdot 53^{4} + 5\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 26\cdot 53 + 39\cdot 53^{2} + 51\cdot 53^{3} + 18\cdot 53^{4} + 6\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 50 + 11\cdot 53 + 17\cdot 53^{2} + 9\cdot 53^{3} + 40\cdot 53^{4} + 30\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
| $(1,5,3,8)(2,7,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,5,3,8)(2,7,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
