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