Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 28\cdot 67 + 27\cdot 67^{2} + 52\cdot 67^{3} + 50\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 24\cdot 67 + 53\cdot 67^{2} + 10\cdot 67^{3} + 27\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 23\cdot 67 + 32\cdot 67^{2} + 56\cdot 67^{3} + 8\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 58\cdot 67 + 20\cdot 67^{2} + 14\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 30\cdot 67 + 51\cdot 67^{2} + 39\cdot 67^{3} + 61\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 + 21\cdot 67 + 29\cdot 67^{2} + 23\cdot 67^{3} + 16\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 64\cdot 67 + 18\cdot 67^{2} + 43\cdot 67^{3} + 14\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 16\cdot 67 + 34\cdot 67^{2} + 27\cdot 67^{3} + 41\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,7,5)(2,6,8,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,3)(4,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,7,5)(2,6,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
