Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 38\cdot 193 + 52\cdot 193^{2} + 128\cdot 193^{3} + 60\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 13\cdot 193 + 90\cdot 193^{2} + 19\cdot 193^{3} + 56\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 87\cdot 193 + 162\cdot 193^{2} + 34\cdot 193^{3} + 26\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 189\cdot 193 + 20\cdot 193^{2} + 106\cdot 193^{3} + 133\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 95 + 96\cdot 193 + 112\cdot 193^{2} + 32\cdot 193^{3} + 170\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 107 + 145\cdot 193 + 29\cdot 193^{2} + 132\cdot 193^{3} + 135\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 122 + 56\cdot 193 + 24\cdot 193^{2} + 89\cdot 193^{3} + 54\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 145 + 145\cdot 193 + 86\cdot 193^{2} + 36\cdot 193^{3} + 135\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,3)(2,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,5,8)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
