Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 52\cdot 89 + 16\cdot 89^{2} + 46\cdot 89^{3} + 50\cdot 89^{4} + 2\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 81\cdot 89 + 25\cdot 89^{2} + 52\cdot 89^{3} + 25\cdot 89^{4} + 25\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 31\cdot 89 + 13\cdot 89^{2} + 86\cdot 89^{3} + 88\cdot 89^{4} + 71\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 25\cdot 89 + 74\cdot 89^{2} + 28\cdot 89^{3} + 74\cdot 89^{4} + 2\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 62 + 13\cdot 89 + 33\cdot 89^{2} + 82\cdot 89^{3} + 12\cdot 89^{4} + 78\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 89 + 70\cdot 89^{2} + 3\cdot 89^{3} + 17\cdot 89^{4} + 46\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 78 + 86\cdot 89 + 53\cdot 89^{2} + 20\cdot 89^{3} + 40\cdot 89^{4} + 5\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 85 + 63\cdot 89 + 68\cdot 89^{2} + 35\cdot 89^{3} + 46\cdot 89^{4} + 34\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,7)(2,6,3,8)$ |
| $(1,2)(3,5)(4,8)(6,7)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,4)(3,7)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,5,7)(2,6,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
