Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 22\cdot 67 + 21\cdot 67^{2} + 22\cdot 67^{3} + 56\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 4\cdot 67 + 3\cdot 67^{2} + 14\cdot 67^{3} + 4\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 49\cdot 67 + 36\cdot 67^{2} + 12\cdot 67^{3} + 38\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 14\cdot 67 + 51\cdot 67^{2} + 19\cdot 67^{3} + 51\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 30\cdot 67 + 51\cdot 67^{2} + 62\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 + 25\cdot 67 + 41\cdot 67^{2} + 51\cdot 67^{3} + 22\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 38 + 33\cdot 67 + 7\cdot 67^{2} + 41\cdot 67^{3} + 58\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 54 + 21\cdot 67 + 55\cdot 67^{2} + 43\cdot 67^{3} + 18\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,4,7)(2,8,5,3)$ |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(1,3,4,8)(2,6,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ |
| $2$ | $4$ | $(1,6,4,7)(2,8,5,3)$ | $0$ |
| $2$ | $4$ | $(1,3,4,8)(2,6,5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,4,2)(3,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
