Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 27\cdot 67 + 47\cdot 67^{2} + 6\cdot 67^{3} + 67^{4} + 23\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 55\cdot 67 + 18\cdot 67^{2} + 9\cdot 67^{3} + 20\cdot 67^{4} + 22\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 47\cdot 67 + 53\cdot 67^{2} + 52\cdot 67^{3} + 65\cdot 67^{4} + 38\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 + 3\cdot 67 + 14\cdot 67^{2} + 65\cdot 67^{3} + 46\cdot 67^{4} + 49\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)$ |
| $(1,3)(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
