Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 29\cdot 53 + 16\cdot 53^{2} + 17\cdot 53^{3} + 42\cdot 53^{4} + 31\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 37\cdot 53 + 19\cdot 53^{2} + 29\cdot 53^{3} + 36\cdot 53^{4} + 15\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 12\cdot 53 + 18\cdot 53^{2} + 44\cdot 53^{3} + 52\cdot 53^{4} + 14\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 20\cdot 53 + 48\cdot 53^{2} + 38\cdot 53^{3} + 41\cdot 53^{4} + 29\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 32\cdot 53 + 4\cdot 53^{2} + 14\cdot 53^{3} + 11\cdot 53^{4} + 23\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 40\cdot 53 + 34\cdot 53^{2} + 8\cdot 53^{3} + 38\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 15\cdot 53 + 33\cdot 53^{2} + 23\cdot 53^{3} + 16\cdot 53^{4} + 37\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 23\cdot 53 + 36\cdot 53^{2} + 35\cdot 53^{3} + 10\cdot 53^{4} + 21\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,2)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
