Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 14\cdot 67 + 8\cdot 67^{2} + 39\cdot 67^{3} + 2\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 19\cdot 67 + 5\cdot 67^{2} + 15\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 31\cdot 67 + 13\cdot 67^{2} + 43\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 10\cdot 67 + 16\cdot 67^{2} + 14\cdot 67^{3} + 14\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 56\cdot 67 + 50\cdot 67^{2} + 52\cdot 67^{3} + 52\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 35\cdot 67 + 53\cdot 67^{2} + 23\cdot 67^{3} + 2\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 47\cdot 67 + 61\cdot 67^{2} + 51\cdot 67^{3} + 2\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 52\cdot 67 + 58\cdot 67^{2} + 27\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
