Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 59\cdot 199 + 14\cdot 199^{2} + 5\cdot 199^{3} + 123\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 + 99\cdot 199 + 46\cdot 199^{2} + 36\cdot 199^{3} + 78\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 87 + 33\cdot 199 + 35\cdot 199^{2} + 103\cdot 199^{3} + 102\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 92 + 38\cdot 199 + 65\cdot 199^{2} + 133\cdot 199^{3} + 164\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 97 + 168\cdot 199 + 110\cdot 199^{2} + 153\cdot 199^{3} + 46\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 129 + 5\cdot 199 + 52\cdot 199^{2} + 147\cdot 199^{3} + 95\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 132 + 164\cdot 199 + 21\cdot 199^{2} + 92\cdot 199^{3} + 132\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 151 + 27\cdot 199 + 52\cdot 199^{2} + 125\cdot 199^{3} + 52\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,4,7,8,6,2)$ |
| $(1,5,7,6)(2,3,4,8)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
| $(1,6)(2,4)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,6)(2,4)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,4,8)$ | $0$ |
| $4$ | $4$ | $(1,3,7,8)(2,6,4,5)$ | $0$ |
| $2$ | $8$ | $(1,3,5,4,7,8,6,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,5,2,7,3,6,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
