Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 123\cdot 199 + 191\cdot 199^{2} + 166\cdot 199^{3} + 117\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 61 + 137\cdot 199 + 135\cdot 199^{2} + 154\cdot 199^{3} + 132\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 107 + 68\cdot 199 + 168\cdot 199^{2} + 100\cdot 199^{3} + 179\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 133 + 55\cdot 199 + 27\cdot 199^{2} + 109\cdot 199^{3} + 189\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 157 + 41\cdot 199 + 43\cdot 199^{2} + 199^{3} + 128\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 167 + 34\cdot 199 + 199^{2} + 81\cdot 199^{3} + 155\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 173 + 174\cdot 199 + 166\cdot 199^{2} + 76\cdot 199^{3} + 185\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 192 + 159\cdot 199 + 61\cdot 199^{2} + 105\cdot 199^{3} + 105\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,8)(3,6)(4,7)$ |
| $(2,8)(3,6)$ |
| $(1,3,5,6)(2,4,8,7)$ |
| $(2,6,8,3)(4,7)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,8)(3,6)(4,7)$ | $-4$ |
| $2$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $2$ | $2$ | $(2,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $4$ | $4$ | $(1,3,5,6)(2,4,8,7)$ | $0$ |
| $4$ | $4$ | $(1,2,7,6)(3,5,8,4)$ | $0$ |
| $4$ | $4$ | $(1,6,7,2)(3,4,8,5)$ | $0$ |
| $4$ | $4$ | $(2,6,8,3)(4,7)$ | $0$ |
| $4$ | $4$ | $(2,3,8,6)(4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
