Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 147\cdot 193 + 110\cdot 193^{2} + 179\cdot 193^{3} + 164\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 43\cdot 193 + 16\cdot 193^{2} + 19\cdot 193^{3} + 164\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 113\cdot 193 + 175\cdot 193^{2} + 18\cdot 193^{3} + 130\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 90 + 179\cdot 193 + 122\cdot 193^{2} + 165\cdot 193^{3} + 99\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 103 + 13\cdot 193 + 70\cdot 193^{2} + 27\cdot 193^{3} + 93\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 138 + 79\cdot 193 + 17\cdot 193^{2} + 174\cdot 193^{3} + 62\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 178 + 149\cdot 193 + 176\cdot 193^{2} + 173\cdot 193^{3} + 28\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 186 + 45\cdot 193 + 82\cdot 193^{2} + 13\cdot 193^{3} + 28\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
