Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45\cdot 67 + 24\cdot 67^{2} + 6\cdot 67^{3} + 28\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 46\cdot 67 + 20\cdot 67^{2} + 41\cdot 67^{3} + 40\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 23\cdot 67 + 65\cdot 67^{2} + 53\cdot 67^{3} + 56\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 52\cdot 67 + 63\cdot 67^{2} + 48\cdot 67^{3} + 65\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 25\cdot 67 + 23\cdot 67^{2} + 58\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 58\cdot 67 + 47\cdot 67^{2} + 15\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 16\cdot 67 + 65\cdot 67^{2} + 27\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 65\cdot 67 + 23\cdot 67^{2} + 15\cdot 67^{3} + 14\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,7,4,2,6,5,3)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,4,2,3)(5,6,7,8)$ |
| $(1,7,2,5)(3,8,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,2,5)(3,8,4,6)$ | $0$ |
| $4$ | $4$ | $(1,4,2,3)(5,6,7,8)$ | $0$ |
| $2$ | $8$ | $(1,3,5,6,2,4,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,5,8,2,3,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
