Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 3\cdot 29 + 29^{2} + 9\cdot 29^{3} + 22\cdot 29^{4} + 22\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 8\cdot 29 + 13\cdot 29^{2} + 24\cdot 29^{3} + 14\cdot 29^{4} + 15\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 7\cdot 29 + 6\cdot 29^{2} + 6\cdot 29^{3} + 13\cdot 29^{4} + 10\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 16\cdot 29 + 10\cdot 29^{2} + 7\cdot 29^{3} + 23\cdot 29^{4} + 25\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 12\cdot 29 + 18\cdot 29^{2} + 21\cdot 29^{3} + 5\cdot 29^{4} + 3\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 + 21\cdot 29 + 22\cdot 29^{2} + 22\cdot 29^{3} + 15\cdot 29^{4} + 18\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 + 20\cdot 29 + 15\cdot 29^{2} + 4\cdot 29^{3} + 14\cdot 29^{4} + 13\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 28 + 25\cdot 29 + 27\cdot 29^{2} + 19\cdot 29^{3} + 6\cdot 29^{4} + 6\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,6)(2,4,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
