Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 44\cdot 67 + 64\cdot 67^{2} + 35\cdot 67^{3} + 23\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 5\cdot 67 + 3\cdot 67^{2} + 48\cdot 67^{3} + 2\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 10\cdot 67 + 56\cdot 67^{2} + 31\cdot 67^{3} + 44\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 18\cdot 67 + 58\cdot 67^{2} + 24\cdot 67^{3} + 46\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 38\cdot 67 + 40\cdot 67^{2} + 32\cdot 67^{3} + 65\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 66\cdot 67 + 45\cdot 67^{2} + 44\cdot 67^{3} + 44\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 4\cdot 67 + 32\cdot 67^{2} + 28\cdot 67^{3} + 19\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 12\cdot 67 + 34\cdot 67^{2} + 21\cdot 67^{3} + 21\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,6)(3,8)(4,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,8,2,7)(3,5,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
