Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 62\cdot 89 + 10\cdot 89^{2} + 13\cdot 89^{3} + 16\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 49\cdot 89 + 3\cdot 89^{2} + 3\cdot 89^{3} + 33\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 8\cdot 89 + 79\cdot 89^{2} + 79\cdot 89^{3} + 24\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 41\cdot 89 + 75\cdot 89^{2} + 32\cdot 89^{3} + 24\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 85\cdot 89 + 5\cdot 89^{2} + 20\cdot 89^{3} + 33\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 30\cdot 89 + 77\cdot 89^{2} + 15\cdot 89^{3} + 83\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 72 + 62\cdot 89 + 72\cdot 89^{2} + 52\cdot 89^{3} + 50\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 15\cdot 89 + 31\cdot 89^{2} + 49\cdot 89^{3} + 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,7,5)(2,3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
