Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 79\cdot 199 + 81\cdot 199^{2} + 83\cdot 199^{3} + 64\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 3\cdot 199 + 163\cdot 199^{2} + 61\cdot 199^{3} + 142\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 127\cdot 199 + 13\cdot 199^{2} + 130\cdot 199^{3} + 68\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 57\cdot 199 + 198\cdot 199^{2} + 172\cdot 199^{3} + 93\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 96 + 146\cdot 199 + 8\cdot 199^{2} + 172\cdot 199^{3} + 2\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 98 + 117\cdot 199 + 82\cdot 199^{2} + 53\cdot 199^{3} + 48\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 115 + 196\cdot 199 + 172\cdot 199^{2} + 12\cdot 199^{3} + 133\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 182 + 68\cdot 199 + 75\cdot 199^{2} + 109\cdot 199^{3} + 43\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,4)(7,8)$ |
| $(1,5,2,6)(3,8,4,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,2,4)(5,7,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,4)(7,8)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,5,2,6)(3,7,4,8)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,6,2,5)(3,8,4,7)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,5,2,6)(3,8,4,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,4)(5,7,6,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,7,2,8)(3,5,4,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
