Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 21\cdot 29 + 15\cdot 29^{2} + 6\cdot 29^{3} + 19\cdot 29^{4} + 10\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 7\cdot 29 + 8\cdot 29^{2} + 13\cdot 29^{3} + 5\cdot 29^{4} + 17\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 19\cdot 29^{2} + 16\cdot 29^{3} + 12\cdot 29^{4} + 17\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 15\cdot 29 + 11\cdot 29^{2} + 23\cdot 29^{3} + 27\cdot 29^{4} + 23\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 13\cdot 29 + 17\cdot 29^{2} + 5\cdot 29^{3} + 29^{4} + 5\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 28\cdot 29 + 9\cdot 29^{2} + 12\cdot 29^{3} + 16\cdot 29^{4} + 11\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 + 21\cdot 29 + 20\cdot 29^{2} + 15\cdot 29^{3} + 23\cdot 29^{4} + 11\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 + 7\cdot 29 + 13\cdot 29^{2} + 22\cdot 29^{3} + 9\cdot 29^{4} + 18\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,4,6)(2,5,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
