Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 10\cdot 67 + 48\cdot 67^{2} + 37\cdot 67^{3} + 14\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 21\cdot 67 + 56\cdot 67^{2} + 13\cdot 67^{3} + 16\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 10\cdot 67 + 4\cdot 67^{2} + 44\cdot 67^{3} + 10\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 47\cdot 67 + 59\cdot 67^{2} + 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 36\cdot 67 + 7\cdot 67^{2} + 31\cdot 67^{3} + 62\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 66\cdot 67 + 21\cdot 67^{2} + 51\cdot 67^{3} + 40\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 + 25\cdot 67 + 25\cdot 67^{2} + 38\cdot 67^{3} + 25\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 50\cdot 67 + 44\cdot 67^{2} + 50\cdot 67^{3} + 29\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,6)(2,7,8,5)$ |
| $(1,2)(3,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,8)(3,6)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,3)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,6)(2,7,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
