Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 48\cdot 89 + 32\cdot 89^{2} + 8\cdot 89^{3} + 12\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 75\cdot 89 + 49\cdot 89^{2} + 11\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 50\cdot 89 + 48\cdot 89^{2} + 78\cdot 89^{3} + 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 51\cdot 89 + 54\cdot 89^{2} + 85\cdot 89^{3} + 33\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 + 37\cdot 89 + 34\cdot 89^{2} + 3\cdot 89^{3} + 55\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 38\cdot 89 + 40\cdot 89^{2} + 10\cdot 89^{3} + 87\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 + 13\cdot 89 + 39\cdot 89^{2} + 77\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 88 + 40\cdot 89 + 56\cdot 89^{2} + 80\cdot 89^{3} + 76\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(1,5,6,2,8,4,3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,6,2,8,4,3,7)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,2,3,5,8,7,6,4)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,4,6,7,8,5,3,2)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,7,3,4,8,2,6,5)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
