Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 171\cdot 199 + 31\cdot 199^{2} + 5\cdot 199^{3} + 43\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 18\cdot 199 + 75\cdot 199^{2} + 38\cdot 199^{3} + 155\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 14\cdot 199 + 50\cdot 199^{2} + 82\cdot 199^{3} + 20\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 188\cdot 199 + 52\cdot 199^{2} + 166\cdot 199^{3} + 96\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 + 168\cdot 199 + 166\cdot 199^{2} + 42\cdot 199^{3} + 51\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 125 + 20\cdot 199 + 39\cdot 199^{2} + 188\cdot 199^{3} + 102\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 132 + 194\cdot 199 + 41\cdot 199^{2} + 73\cdot 199^{3} + 179\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 160 + 20\cdot 199 + 139\cdot 199^{2} + 147\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,6)(5,8)$ |
| $(1,3,2,7)(4,8,6,5)$ |
| $(1,5)(2,8)(3,4)(6,7)$ |
| $(1,3,2,7)(4,5,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(4,6)(5,8)$ | $0$ |
| $1$ | $4$ | $(1,3,2,7)(4,8,6,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,2,3)(4,5,6,8)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,2,7)(4,5,6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,2,8)(3,4,7,6)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
