Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 36\cdot 67 + 39\cdot 67^{2} + 53\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 54\cdot 67 + 7\cdot 67^{2} + 5\cdot 67^{3} + 31\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 7\cdot 67 + 57\cdot 67^{2} + 6\cdot 67^{3} + 40\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 3\cdot 67 + 29\cdot 67^{2} + 21\cdot 67^{3} + 28\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 24\cdot 67 + 67^{2} + 37\cdot 67^{3} + 43\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 28\cdot 67 + 13\cdot 67^{2} + 7\cdot 67^{3} + 30\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 42 + 9\cdot 67 + 7\cdot 67^{2} + 51\cdot 67^{3} + 32\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 37\cdot 67 + 45\cdot 67^{2} + 18\cdot 67^{3} + 44\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2,4,8)(3,6,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,8)(3,5)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,5)(3,8)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,8)(3,6,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
