Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 2\cdot 29 + 19\cdot 29^{2} + 11\cdot 29^{3} + 10\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 2\cdot 29 + 22\cdot 29^{2} + 18\cdot 29^{3} + 10\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 20\cdot 29^{2} + 27\cdot 29^{3} + 28\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 4\cdot 29 + 25\cdot 29^{2} + 8\cdot 29^{3} + 14\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 + 26\cdot 29 + 18\cdot 29^{2} + 17\cdot 29^{3} + 27\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 + 6\cdot 29 + 22\cdot 29^{2} + 10\cdot 29^{3} + 23\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 + 13\cdot 29 + 15\cdot 29^{2} + 15\cdot 29^{3} + 14\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 + 2\cdot 29 + 2\cdot 29^{2} + 5\cdot 29^{3} + 15\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,4)(3,5)(6,8)$ |
| $(1,3,6,4)(2,7,5,8)$ |
| $(2,5)(7,8)$ |
| $(1,6)(2,5)(3,4)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(2,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,3,6,4)(2,7,5,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,6,3)(2,8,5,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,6,2)(3,8,4,7)$ | $0$ |
| $2$ | $4$ | $(1,8,6,7)(2,4,5,3)$ | $0$ |
| $2$ | $4$ | $(1,3,6,4)(2,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
