Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 89 + 61\cdot 89^{2} + 45\cdot 89^{3} + 57\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 85\cdot 89 + 29\cdot 89^{2} + 38\cdot 89^{3} + 54\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 33\cdot 89 + 39\cdot 89^{2} + 84\cdot 89^{3} + 81\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 28\cdot 89 + 8\cdot 89^{2} + 77\cdot 89^{3} + 78\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 + 60\cdot 89 + 80\cdot 89^{2} + 11\cdot 89^{3} + 10\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 + 55\cdot 89 + 49\cdot 89^{2} + 4\cdot 89^{3} + 7\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 78 + 3\cdot 89 + 59\cdot 89^{2} + 50\cdot 89^{3} + 34\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 87 + 87\cdot 89 + 27\cdot 89^{2} + 43\cdot 89^{3} + 31\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,7,6)(2,3,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
