Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 30\cdot 67 + 27\cdot 67^{2} + 19\cdot 67^{3} + 20\cdot 67^{4} + 60\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 21\cdot 67 + 18\cdot 67^{2} + 35\cdot 67^{3} + 52\cdot 67^{4} + 18\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 12\cdot 67 + 66\cdot 67^{2} + 15\cdot 67^{3} + 16\cdot 67^{4} + 60\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 11\cdot 67 + 38\cdot 67^{2} + 57\cdot 67^{3} + 59\cdot 67^{4} + 5\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 12\cdot 67 + 45\cdot 67^{2} + 4\cdot 67^{3} + 52\cdot 67^{4} + 13\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 + 26\cdot 67 + 58\cdot 67^{2} + 3\cdot 67^{3} + 10\cdot 67^{4} + 12\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 48 + 13\cdot 67 + 48\cdot 67^{2} + 37\cdot 67^{3} + 27\cdot 67^{4} + 64\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 5\cdot 67 + 33\cdot 67^{2} + 26\cdot 67^{3} + 29\cdot 67^{4} + 32\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,8)(5,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,6,8,3,2,5,7,4)$ |
| $(1,2)(7,8)$ |
| $(1,8,2,7)(3,6,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,2)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,8,2,7)(3,5,4,6)$ | $0$ |
| $2$ | $4$ | $(1,8,2,7)(3,6,4,5)$ | $0$ |
| $4$ | $4$ | $(1,4,2,3)(5,7,6,8)$ | $0$ |
| $4$ | $8$ | $(1,6,8,3,2,5,7,4)$ | $0$ |
| $4$ | $8$ | $(1,6,7,4,2,5,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
