Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 20\cdot 29 + 25\cdot 29^{2} + 17\cdot 29^{3} + 9\cdot 29^{4} + 9\cdot 29^{5} + 11\cdot 29^{6} + 12\cdot 29^{7} + 5\cdot 29^{8} + 9\cdot 29^{9} +O\left(29^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 23\cdot 29 + 27\cdot 29^{2} + 20\cdot 29^{3} + 28\cdot 29^{4} + 5\cdot 29^{5} + 18\cdot 29^{6} + 18\cdot 29^{7} + 11\cdot 29^{8} + 2\cdot 29^{9} + 29^{10} +O\left(29^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 26\cdot 29 + 17\cdot 29^{2} + 22\cdot 29^{3} + 15\cdot 29^{4} + 19\cdot 29^{5} + 4\cdot 29^{6} + 25\cdot 29^{7} + 5\cdot 29^{8} +O\left(29^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 + 4\cdot 29 + 25\cdot 29^{2} + 7\cdot 29^{3} + 28\cdot 29^{4} + 10\cdot 29^{5} + 17\cdot 29^{6} + 10\cdot 29^{7} + 5\cdot 29^{8} + 2\cdot 29^{10} +O\left(29^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 24\cdot 29 + 3\cdot 29^{2} + 21\cdot 29^{3} + 18\cdot 29^{5} + 11\cdot 29^{6} + 18\cdot 29^{7} + 23\cdot 29^{8} + 28\cdot 29^{9} + 26\cdot 29^{10} +O\left(29^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 2\cdot 29 + 11\cdot 29^{2} + 6\cdot 29^{3} + 13\cdot 29^{4} + 9\cdot 29^{5} + 24\cdot 29^{6} + 3\cdot 29^{7} + 23\cdot 29^{8} + 28\cdot 29^{9} + 28\cdot 29^{10} +O\left(29^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 + 5\cdot 29 + 29^{2} + 8\cdot 29^{3} + 23\cdot 29^{5} + 10\cdot 29^{6} + 10\cdot 29^{7} + 17\cdot 29^{8} + 26\cdot 29^{9} + 27\cdot 29^{10} +O\left(29^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 + 8\cdot 29 + 3\cdot 29^{2} + 11\cdot 29^{3} + 19\cdot 29^{4} + 19\cdot 29^{5} + 17\cdot 29^{6} + 16\cdot 29^{7} + 23\cdot 29^{8} + 19\cdot 29^{9} + 28\cdot 29^{10} +O\left(29^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(2,4,7,5)$ |
| $(2,7)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(2,4,7,5)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(2,5,7,4)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3,8,6)(2,7)(4,5)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,3)(2,7)(4,5)$ | $-\zeta_{4} + 1$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,7,3,4,8,2,6,5)$ | $0$ |
| $4$ | $8$ | $(1,4,6,7,8,5,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
