Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 44\cdot 89 + 66\cdot 89^{2} + 63\cdot 89^{3} + 79\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 32\cdot 89 + 66\cdot 89^{2} + 80\cdot 89^{3} + 23\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 41\cdot 89 + 29\cdot 89^{2} + 80\cdot 89^{3} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 42\cdot 89 + 19\cdot 89^{2} + 25\cdot 89^{3} + 49\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 39\cdot 89 + 45\cdot 89^{2} + 6\cdot 89^{3} + 28\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 62\cdot 89 + 62\cdot 89^{2} + 80\cdot 89^{3} + 14\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 54 + 14\cdot 89 + 70\cdot 89^{2} + 13\cdot 89^{3} + 2\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 80 + 79\cdot 89 + 84\cdot 89^{2} + 4\cdot 89^{3} + 68\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,6)(3,4)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,7)(2,3,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
