Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 29\cdot 53 + 48\cdot 53^{2} + 2\cdot 53^{3} + 24\cdot 53^{4} + 3\cdot 53^{5} + 43\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 25\cdot 53 + 3\cdot 53^{2} + 34\cdot 53^{3} + 45\cdot 53^{4} + 7\cdot 53^{5} + 3\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 10\cdot 53 + 38\cdot 53^{2} + 22\cdot 53^{3} + 51\cdot 53^{4} + 5\cdot 53^{5} + 20\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 52\cdot 53 + 26\cdot 53^{2} + 38\cdot 53^{3} + 50\cdot 53^{4} + 45\cdot 53^{5} + 29\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 26\cdot 53^{2} + 14\cdot 53^{3} + 2\cdot 53^{4} + 7\cdot 53^{5} + 23\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 42\cdot 53 + 14\cdot 53^{2} + 30\cdot 53^{3} + 53^{4} + 47\cdot 53^{5} + 32\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 + 27\cdot 53 + 49\cdot 53^{2} + 18\cdot 53^{3} + 7\cdot 53^{4} + 45\cdot 53^{5} + 49\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 48 + 23\cdot 53 + 4\cdot 53^{2} + 50\cdot 53^{3} + 28\cdot 53^{4} + 49\cdot 53^{5} + 9\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,2)(4,5)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
