Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 11\cdot 67 + 14\cdot 67^{2} + 46\cdot 67^{3} + 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 45\cdot 67 + 12\cdot 67^{2} + 9\cdot 67^{3} + 60\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 53\cdot 67 + 40\cdot 67^{2} + 52\cdot 67^{3} + 52\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 60\cdot 67 + 14\cdot 67^{2} + 25\cdot 67^{3} + 18\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 58\cdot 67 + 21\cdot 67^{2} + 66\cdot 67^{3} + 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 9\cdot 67 + 64\cdot 67^{2} + 9\cdot 67^{3} + 61\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 5\cdot 67 + 33\cdot 67^{2} + 32\cdot 67^{3} + 52\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 53 + 24\cdot 67 + 66\cdot 67^{2} + 25\cdot 67^{3} + 19\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,5)(6,7)$ |
| $(1,3)(2,6)(4,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,6,5,8)(2,3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
