Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 164\cdot 199 + 198\cdot 199^{2} + 81\cdot 199^{3} + 38\cdot 199^{4} + 171\cdot 199^{5} + 19\cdot 199^{6} + 147\cdot 199^{7} + 174\cdot 199^{8} + 155\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 63 + 35\cdot 199 + 68\cdot 199^{2} + 128\cdot 199^{3} + 180\cdot 199^{4} + 106\cdot 199^{5} + 115\cdot 199^{6} + 61\cdot 199^{7} + 198\cdot 199^{8} + 48\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 87 + 123\cdot 199 + 23\cdot 199^{2} + 9\cdot 199^{3} + 165\cdot 199^{4} + 77\cdot 199^{5} + 53\cdot 199^{6} + 110\cdot 199^{7} + 20\cdot 199^{8} + 26\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 92 + 157\cdot 199 + 9\cdot 199^{2} + 102\cdot 199^{3} + 25\cdot 199^{4} + 88\cdot 199^{5} + 24\cdot 199^{6} + 22\cdot 199^{7} + 4\cdot 199^{8} + 42\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 107 + 41\cdot 199 + 189\cdot 199^{2} + 96\cdot 199^{3} + 173\cdot 199^{4} + 110\cdot 199^{5} + 174\cdot 199^{6} + 176\cdot 199^{7} + 194\cdot 199^{8} + 156\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 112 + 75\cdot 199 + 175\cdot 199^{2} + 189\cdot 199^{3} + 33\cdot 199^{4} + 121\cdot 199^{5} + 145\cdot 199^{6} + 88\cdot 199^{7} + 178\cdot 199^{8} + 172\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 136 + 163\cdot 199 + 130\cdot 199^{2} + 70\cdot 199^{3} + 18\cdot 199^{4} + 92\cdot 199^{5} + 83\cdot 199^{6} + 137\cdot 199^{7} + 150\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 158 + 34\cdot 199 + 117\cdot 199^{3} + 160\cdot 199^{4} + 27\cdot 199^{5} + 179\cdot 199^{6} + 51\cdot 199^{7} + 24\cdot 199^{8} + 43\cdot 199^{9} +O\left(199^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,7)(4,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,4,7,8,6,5,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
