Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 47\cdot 67 + 48\cdot 67^{2} + 56\cdot 67^{3} + 53\cdot 67^{4} + 51\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 20\cdot 67 + 18\cdot 67^{2} + 32\cdot 67^{3} + 41\cdot 67^{4} + 34\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 2\cdot 67 + 29\cdot 67^{2} + 50\cdot 67^{3} + 17\cdot 67^{4} + 54\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 13\cdot 67 + 10\cdot 67^{2} + 6\cdot 67^{3} + 33\cdot 67^{4} + 11\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 53\cdot 67 + 56\cdot 67^{2} + 60\cdot 67^{3} + 33\cdot 67^{4} + 55\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 64\cdot 67 + 37\cdot 67^{2} + 16\cdot 67^{3} + 49\cdot 67^{4} + 12\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 46\cdot 67 + 48\cdot 67^{2} + 34\cdot 67^{3} + 25\cdot 67^{4} + 32\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 19\cdot 67 + 18\cdot 67^{2} + 10\cdot 67^{3} + 13\cdot 67^{4} + 15\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
