Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 178\cdot 199 + 91\cdot 199^{2} + 76\cdot 199^{4} + 116\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 85\cdot 199 + 99\cdot 199^{2} + 166\cdot 199^{3} + 117\cdot 199^{4} + 65\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 26\cdot 199 + 109\cdot 199^{2} + 169\cdot 199^{3} + 196\cdot 199^{4} + 103\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 + 188\cdot 199 + 177\cdot 199^{2} + 126\cdot 199^{3} + 130\cdot 199^{4} + 158\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 146 + 10\cdot 199 + 21\cdot 199^{2} + 72\cdot 199^{3} + 68\cdot 199^{4} + 40\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 156 + 172\cdot 199 + 89\cdot 199^{2} + 29\cdot 199^{3} + 2\cdot 199^{4} + 95\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 168 + 113\cdot 199 + 99\cdot 199^{2} + 32\cdot 199^{3} + 81\cdot 199^{4} + 133\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 180 + 20\cdot 199 + 107\cdot 199^{2} + 198\cdot 199^{3} + 122\cdot 199^{4} + 82\cdot 199^{5} +O\left(199^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,7,3,8,5,2,6)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(3,6)(4,5)$ |
| $(1,8)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $0$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $0$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
