Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 123\cdot 193 + 30\cdot 193^{2} + 34\cdot 193^{3} + 37\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 95\cdot 193 + 132\cdot 193^{2} + 146\cdot 193^{3} + 139\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 84 + 120\cdot 193 + 39\cdot 193^{2} + 6\cdot 193^{3} + 157\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 96 + 100\cdot 193 + 51\cdot 193^{2} + 74\cdot 193^{3} + 126\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 97 + 92\cdot 193 + 141\cdot 193^{2} + 118\cdot 193^{3} + 66\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 109 + 72\cdot 193 + 153\cdot 193^{2} + 186\cdot 193^{3} + 35\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 173 + 97\cdot 193 + 60\cdot 193^{2} + 46\cdot 193^{3} + 53\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 186 + 69\cdot 193 + 162\cdot 193^{2} + 158\cdot 193^{3} + 155\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,6)(2,4,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
