Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 7\cdot 67 + 61\cdot 67^{2} + 65\cdot 67^{3} + 58\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 10\cdot 67 + 7\cdot 67^{2} + 12\cdot 67^{3} + 11\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 19\cdot 67 + 22\cdot 67^{3} + 21\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 64\cdot 67 + 42\cdot 67^{2} + 13\cdot 67^{3} + 24\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 60\cdot 67^{2} + 34\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 38\cdot 67 + 5\cdot 67^{2} + 40\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 38\cdot 67 + 35\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 61 + 21\cdot 67 + 23\cdot 67^{2} + 44\cdot 67^{3} + 32\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,6)(5,8)$ |
| $(1,5,7,6,2,8,3,4)$ |
| $(1,7,2,3)(4,5,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $-1$ |
| $1$ | $4$ | $(1,7,2,3)(4,5,6,8)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,3,2,7)(4,8,6,5)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,7,6,2,8,3,4)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,6,3,5,2,4,7,8)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,8,7,4,2,5,3,6)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,4,3,8,2,6,7,5)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
