Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 16\cdot 29 + 15\cdot 29^{2} + 2\cdot 29^{3} + 7\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 26\cdot 29^{2} + 9\cdot 29^{3} + 19\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 + 12\cdot 29 + 12\cdot 29^{2} + 29^{3} + 17\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 13\cdot 29 + 21\cdot 29^{2} + 18\cdot 29^{3} + 3\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 + 24\cdot 29 + 25\cdot 29^{2} + 21\cdot 29^{3} + 17\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 + 11\cdot 29 + 7\cdot 29^{2} + 26\cdot 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 14 + 25\cdot 29 + 14\cdot 29^{2} + 24\cdot 29^{3} + 27\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 16 + 13\cdot 29 + 21\cdot 29^{2} + 10\cdot 29^{3} + 6\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,4)(6,8)$ |
| $(1,7,5,2)(3,8,4,6)$ |
| $(1,6,7,3,5,8,2,4)$ |
| $(1,5)(2,7)(3,4)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $-2$ |
| $2$ | $2$ | $(3,4)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,7,5,2)(3,8,4,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,5,7)(3,6,4,8)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,5,2)(3,6,4,8)$ | $0$ |
| $2$ | $8$ | $(1,6,7,3,5,8,2,4)$ | $0$ |
| $2$ | $8$ | $(1,3,2,6,5,4,7,8)$ | $0$ |
| $2$ | $8$ | $(1,6,2,4,5,8,7,3)$ | $0$ |
| $2$ | $8$ | $(1,4,7,6,5,3,2,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
