Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 67 + 55\cdot 67^{3} + 42\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 + 12\cdot 67 + 50\cdot 67^{2} + 59\cdot 67^{3} + 16\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 52\cdot 67 + 16\cdot 67^{2} + 19\cdot 67^{3} + 7\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $3$ |
$2$ |
$(1,2)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
