Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 20\cdot 67 + 55\cdot 67^{2} + 61\cdot 67^{3} + 46\cdot 67^{4} + 56\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 53\cdot 67 + 47\cdot 67^{2} + 14\cdot 67^{3} + 9\cdot 67^{4} + 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 62\cdot 67 + 53\cdot 67^{2} + 7\cdot 67^{3} + 11\cdot 67^{4} + 55\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 21\cdot 67 + 25\cdot 67^{2} + 32\cdot 67^{3} + 11\cdot 67^{4} + 22\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 42\cdot 67 + 18\cdot 67^{2} + 17\cdot 67^{3} + 55\cdot 67^{4} + 65\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
