Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 23\cdot 67 + 24\cdot 67^{2} + 22\cdot 67^{3} + 29\cdot 67^{4} + 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 46\cdot 67 + 20\cdot 67^{2} + 60\cdot 67^{3} + 26\cdot 67^{4} + 57\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 14\cdot 67 + 48\cdot 67^{2} + 44\cdot 67^{3} + 5\cdot 67^{4} + 51\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 50\cdot 67 + 40\cdot 67^{2} + 6\cdot 67^{3} + 5\cdot 67^{4} + 24\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 16\cdot 67 + 26\cdot 67^{2} + 60\cdot 67^{3} + 61\cdot 67^{4} + 42\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 43 + 52\cdot 67 + 18\cdot 67^{2} + 22\cdot 67^{3} + 61\cdot 67^{4} + 15\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 20\cdot 67 + 46\cdot 67^{2} + 6\cdot 67^{3} + 40\cdot 67^{4} + 9\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 43\cdot 67 + 42\cdot 67^{2} + 44\cdot 67^{3} + 37\cdot 67^{4} + 65\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,3)(5,6,8,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
