Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 26\cdot 199 + 82\cdot 199^{2} + 108\cdot 199^{3} + 19\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 + 121\cdot 199 + 184\cdot 199^{2} + 111\cdot 199^{3} + 144\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 81 + 58\cdot 199 + 78\cdot 199^{2} + 72\cdot 199^{3} + 38\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 83 + 183\cdot 199 + 194\cdot 199^{2} + 33\cdot 199^{3} + 98\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 99 + 65\cdot 199 + 190\cdot 199^{2} + 111\cdot 199^{3} + 125\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 121 + 125\cdot 199 + 139\cdot 199^{2} + 78\cdot 199^{3} + 156\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 158 + 126\cdot 199 + 63\cdot 199^{2} + 76\cdot 199^{3} + 181\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 184 + 88\cdot 199 + 61\cdot 199^{2} + 3\cdot 199^{3} + 32\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,8)(3,5)(4,6)$ |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,4,7,6)(2,5,3,8)$ |
| $(1,7)(2,3)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,5,3,8)$ | $0$ |
| $2$ | $8$ | $(1,8,6,3,7,5,4,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,4,8,7,2,6,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
