Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 31\cdot 89 + 4\cdot 89^{2} + 47\cdot 89^{3} + 68\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 35\cdot 89 + 20\cdot 89^{2} + 74\cdot 89^{3} + 54\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 7\cdot 89 + 6\cdot 89^{2} + 64\cdot 89^{3} + 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 68\cdot 89 + 15\cdot 89^{2} + 29\cdot 89^{3} + 87\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 + 18\cdot 89 + 16\cdot 89^{2} + 82\cdot 89^{3} + 81\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 + 20\cdot 89 + 30\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 57\cdot 89 + 68\cdot 89^{2} + 71\cdot 89^{3} + 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 76 + 27\cdot 89 + 46\cdot 89^{2} + 46\cdot 89^{3} + 39\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,4,7,6)(2,5,3,8)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,5,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
