Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 194\cdot 199 + 117\cdot 199^{2} + 84\cdot 199^{3} + 171\cdot 199^{4} + 187\cdot 199^{5} + 124\cdot 199^{6} + 102\cdot 199^{7} + 145\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 190\cdot 199 + 79\cdot 199^{2} + 133\cdot 199^{3} + 51\cdot 199^{4} + 117\cdot 199^{5} + 129\cdot 199^{6} + 88\cdot 199^{7} + 49\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 149\cdot 199 + 144\cdot 199^{2} + 109\cdot 199^{3} + 19\cdot 199^{4} + 188\cdot 199^{5} + 5\cdot 199^{6} + 184\cdot 199^{7} + 82\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 81 + 119\cdot 199 + 163\cdot 199^{2} + 55\cdot 199^{3} + 75\cdot 199^{4} + 137\cdot 199^{5} + 39\cdot 199^{6} + 153\cdot 199^{7} + 128\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 118 + 79\cdot 199 + 35\cdot 199^{2} + 143\cdot 199^{3} + 123\cdot 199^{4} + 61\cdot 199^{5} + 159\cdot 199^{6} + 45\cdot 199^{7} + 70\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 179 + 49\cdot 199 + 54\cdot 199^{2} + 89\cdot 199^{3} + 179\cdot 199^{4} + 10\cdot 199^{5} + 193\cdot 199^{6} + 14\cdot 199^{7} + 116\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 184 + 8\cdot 199 + 119\cdot 199^{2} + 65\cdot 199^{3} + 147\cdot 199^{4} + 81\cdot 199^{5} + 69\cdot 199^{6} + 110\cdot 199^{7} + 149\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 186 + 4\cdot 199 + 81\cdot 199^{2} + 114\cdot 199^{3} + 27\cdot 199^{4} + 11\cdot 199^{5} + 74\cdot 199^{6} + 96\cdot 199^{7} + 53\cdot 199^{8} +O\left(199^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
| $(1,8)(2,7)$ |
| $(1,2,8,7)(3,4,6,5)$ |
| $(1,3,7,4,8,6,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $0$ |
| $4$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
