Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 20\cdot 53 + 38\cdot 53^{2} + 51\cdot 53^{3} + 44\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 20\cdot 53 + 11\cdot 53^{2} + 34\cdot 53^{3} + 22\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 3\cdot 53 + 28\cdot 53^{2} + 51\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 40\cdot 53 + 53^{2} + 20\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 45\cdot 53 + 52\cdot 53^{2} + 7\cdot 53^{3} + 29\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 52\cdot 53 + 39\cdot 53^{2} + 43\cdot 53^{3} + 45\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 11\cdot 53 + 45\cdot 53^{2} + 49\cdot 53^{3} + 6\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 17\cdot 53 + 47\cdot 53^{2} + 3\cdot 53^{3} + 3\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,8,2,3,4,7,5)$ |
| $(2,6)(4,5)(7,8)$ |
| $(1,7,3,8)(2,6,5,4)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)(7,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(2,6)(4,5)(7,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,3,7)(2,4,5,6)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,2,3,5)(4,8,6,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,8,2,3,4,7,5)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,4,8,5,3,6,7,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
