Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 132\cdot 193 + 146\cdot 193^{2} + 104\cdot 193^{3} + 75\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 78 + 161\cdot 193 + 142\cdot 193^{2} + 96\cdot 193^{3} + 6\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 84 + 50\cdot 193 + 121\cdot 193^{2} + 136\cdot 193^{3} + 190\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 95 + 64\cdot 193 + 147\cdot 193^{2} + 7\cdot 193^{3} + 24\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 130 + 109\cdot 193 + 167\cdot 193^{2} + 144\cdot 193^{3} + 164\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 173 + 27\cdot 193 + 142\cdot 193^{2} + 176\cdot 193^{3} + 86\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 182 + 183\cdot 193 + 121\cdot 193^{2} + 56\cdot 193^{3} + 110\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 184 + 41\cdot 193 + 168\cdot 193^{2} + 47\cdot 193^{3} + 113\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,6)(3,7)(5,8)$ |
| $(1,2,3,8)(4,5,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,3,8)(4,5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
