Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 23\cdot 67 + 46\cdot 67^{2} + 26\cdot 67^{3} + 26\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 + 56\cdot 67 + 56\cdot 67^{2} + 52\cdot 67^{3} + 56\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 27\cdot 67 + 11\cdot 67^{2} + 25\cdot 67^{3} + 13\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 41\cdot 67 + 44\cdot 67^{2} + 38\cdot 67^{3} + 66\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 61\cdot 67 + 3\cdot 67^{2} + 15\cdot 67^{3} + 42\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 + 40\cdot 67 + 35\cdot 67^{2} + 38\cdot 67^{3} + 50\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 48 + 34\cdot 67 + 21\cdot 67^{2} + 30\cdot 67^{3} + 12\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 48\cdot 67 + 47\cdot 67^{2} + 40\cdot 67^{3} + 66\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,7,3)(4,8,5,6)$ |
| $(1,5,2,6,7,4,3,8)$ |
| $(1,7)(2,3)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $-1$ |
| $1$ | $4$ | $(1,2,7,3)(4,8,5,6)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,3,7,2)(4,6,5,8)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,2,6,7,4,3,8)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,6,3,5,7,8,2,4)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,2,8,7,5,3,6)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,8,3,4,7,6,2,5)$ | $-\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
