Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 15.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 28\cdot 29 + 2\cdot 29^{2} + 27\cdot 29^{3} + 6\cdot 29^{4} + 20\cdot 29^{5} + 19\cdot 29^{6} + 2\cdot 29^{7} + 2\cdot 29^{8} + 15\cdot 29^{9} + 26\cdot 29^{10} + 11\cdot 29^{11} + 4\cdot 29^{12} + 24\cdot 29^{13} + 17\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 7\cdot 29 + 10\cdot 29^{2} + 27\cdot 29^{3} + 20\cdot 29^{4} + 13\cdot 29^{5} + 6\cdot 29^{6} + 11\cdot 29^{7} + 11\cdot 29^{8} + 8\cdot 29^{9} + 25\cdot 29^{10} + 21\cdot 29^{11} + 8\cdot 29^{12} + 28\cdot 29^{13} + 9\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 29 + 21\cdot 29^{2} + 20\cdot 29^{3} + 27\cdot 29^{4} + 3\cdot 29^{5} + 17\cdot 29^{6} + 25\cdot 29^{7} + 8\cdot 29^{8} + 20\cdot 29^{9} + 21\cdot 29^{10} + 28\cdot 29^{11} + 8\cdot 29^{12} + 16\cdot 29^{13} + 24\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 18\cdot 29 + 17\cdot 29^{2} + 27\cdot 29^{3} + 17\cdot 29^{4} + 25\cdot 29^{5} + 13\cdot 29^{6} + 13\cdot 29^{7} + 23\cdot 29^{8} + 9\cdot 29^{9} + 6\cdot 29^{10} + 9\cdot 29^{12} + 4\cdot 29^{13} + 17\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 10\cdot 29 + 11\cdot 29^{2} + 29^{3} + 11\cdot 29^{4} + 3\cdot 29^{5} + 15\cdot 29^{6} + 15\cdot 29^{7} + 5\cdot 29^{8} + 19\cdot 29^{9} + 22\cdot 29^{10} + 28\cdot 29^{11} + 19\cdot 29^{12} + 24\cdot 29^{13} + 11\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 27\cdot 29 + 7\cdot 29^{2} + 8\cdot 29^{3} + 29^{4} + 25\cdot 29^{5} + 11\cdot 29^{6} + 3\cdot 29^{7} + 20\cdot 29^{8} + 8\cdot 29^{9} + 7\cdot 29^{10} + 20\cdot 29^{12} + 12\cdot 29^{13} + 4\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 + 21\cdot 29 + 18\cdot 29^{2} + 29^{3} + 8\cdot 29^{4} + 15\cdot 29^{5} + 22\cdot 29^{6} + 17\cdot 29^{7} + 17\cdot 29^{8} + 20\cdot 29^{9} + 3\cdot 29^{10} + 7\cdot 29^{11} + 20\cdot 29^{12} + 19\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 28 + 26\cdot 29^{2} + 29^{3} + 22\cdot 29^{4} + 8\cdot 29^{5} + 9\cdot 29^{6} + 26\cdot 29^{7} + 26\cdot 29^{8} + 13\cdot 29^{9} + 2\cdot 29^{10} + 17\cdot 29^{11} + 24\cdot 29^{12} + 4\cdot 29^{13} + 11\cdot 29^{14} +O\left(29^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(5,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,4)(2,7)(5,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,4,3,8,7,5,6)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,7,4,6,8,2,5,3)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
