Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 39\cdot 67 + 33\cdot 67^{2} + 31\cdot 67^{3} + 53\cdot 67^{4} + 24\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 16\cdot 67 + 2\cdot 67^{2} + 62\cdot 67^{3} + 55\cdot 67^{4} + 13\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 33\cdot 67 + 21\cdot 67^{2} + 39\cdot 67^{3} + 13\cdot 67^{4} + 46\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 38\cdot 67 + 65\cdot 67^{2} + 17\cdot 67^{3} + 41\cdot 67^{4} + 33\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 28\cdot 67 + 67^{2} + 49\cdot 67^{3} + 25\cdot 67^{4} + 33\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 + 33\cdot 67 + 45\cdot 67^{2} + 27\cdot 67^{3} + 53\cdot 67^{4} + 20\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 50\cdot 67 + 64\cdot 67^{2} + 4\cdot 67^{3} + 11\cdot 67^{4} + 53\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 63 + 27\cdot 67 + 33\cdot 67^{2} + 35\cdot 67^{3} + 13\cdot 67^{4} + 42\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
