Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 134\cdot 199 + 31\cdot 199^{2} + 57\cdot 199^{3} + 122\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 23\cdot 199 + 59\cdot 199^{2} + 50\cdot 199^{3} + 12\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 198\cdot 199 + 17\cdot 199^{2} + 77\cdot 199^{3} + 38\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 82 + 146\cdot 199 + 52\cdot 199^{2} + 176\cdot 199^{3} + 38\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 111 + 195\cdot 199 + 198\cdot 199^{2} + 38\cdot 199^{3} + 116\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 175 + 56\cdot 199 + 128\cdot 199^{2} + 105\cdot 199^{3} + 5\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 182 + 125\cdot 199 + 127\cdot 199^{2} + 195\cdot 199^{3} + 186\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 193 + 114\cdot 199 + 179\cdot 199^{2} + 94\cdot 199^{3} + 76\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,8)(3,6)(4,5)$ |
| $(1,6,7,3)(2,5,8,4)$ |
| $(1,2)(4,5)(7,8)$ |
| $(1,8,7,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,8,7,2)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,6,7,3)(2,5,8,4)$ | $0$ |
| $2$ | $8$ | $(1,6,8,5,7,3,2,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,8,4,7,6,2,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
