Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 20\cdot 89 + 72\cdot 89^{2} + 18\cdot 89^{3} + 36\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 24\cdot 89 + 2\cdot 89^{2} + 58\cdot 89^{3} + 4\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 58\cdot 89 + 26\cdot 89^{2} + 26\cdot 89^{3} + 30\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 + 86\cdot 89 + 69\cdot 89^{2} + 50\cdot 89^{3} + 4\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 2\cdot 89 + 19\cdot 89^{2} + 38\cdot 89^{3} + 84\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 + 30\cdot 89 + 62\cdot 89^{2} + 62\cdot 89^{3} + 58\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 + 64\cdot 89 + 86\cdot 89^{2} + 30\cdot 89^{3} + 84\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 67 + 68\cdot 89 + 16\cdot 89^{2} + 70\cdot 89^{3} + 52\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,5)(4,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
