Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 50\cdot 67 + 26\cdot 67^{2} + 17\cdot 67^{3} + 13\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 66\cdot 67 + 5\cdot 67^{2} + 66\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 43\cdot 67 + 50\cdot 67^{2} + 51\cdot 67^{3} + 10\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 57\cdot 67 + 57\cdot 67^{2} + 41\cdot 67^{3} + 2\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 23\cdot 67 + 45\cdot 67^{2} + 55\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 + 56\cdot 67 + 42\cdot 67^{2} + 26\cdot 67^{3} + 53\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 + 25\cdot 67 + 51\cdot 67^{2} + 48\cdot 67^{3} + 22\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 10\cdot 67 + 54\cdot 67^{2} + 25\cdot 67^{3} + 14\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,7,6)(2,8,3,5)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,3)(2,7)(4,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,8,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
