Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 61\cdot 67 + 9\cdot 67^{2} + 13\cdot 67^{3} + 59\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 19\cdot 67 + 63\cdot 67^{2} + 54\cdot 67^{3} + 54\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 39\cdot 67 + 23\cdot 67^{2} + 23\cdot 67^{3} + 61\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 18\cdot 67 + 32\cdot 67^{3} + 46\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 27\cdot 67 + 22\cdot 67^{2} + 45\cdot 67^{3} + 60\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 + 23\cdot 67 + 25\cdot 67^{2} + 56\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 19 + 16\cdot 67 + 29\cdot 67^{2} + 9\cdot 67^{3} + 12\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 42 + 63\cdot 67 + 26\cdot 67^{2} + 22\cdot 67^{3} + 51\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,7)(5,8)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
