Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 27\cdot 89 + 73\cdot 89^{2} + 14\cdot 89^{3} + 67\cdot 89^{4} + 34\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 33\cdot 89 + 56\cdot 89^{2} + 8\cdot 89^{3} + 68\cdot 89^{4} + 16\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 55\cdot 89 + 38\cdot 89^{2} + 47\cdot 89^{3} + 73\cdot 89^{4} + 3\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 61\cdot 89 + 21\cdot 89^{2} + 41\cdot 89^{3} + 74\cdot 89^{4} + 74\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 27\cdot 89 + 67\cdot 89^{2} + 47\cdot 89^{3} + 14\cdot 89^{4} + 14\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 33\cdot 89 + 50\cdot 89^{2} + 41\cdot 89^{3} + 15\cdot 89^{4} + 85\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 70 + 55\cdot 89 + 32\cdot 89^{2} + 80\cdot 89^{3} + 20\cdot 89^{4} + 72\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 76 + 61\cdot 89 + 15\cdot 89^{2} + 74\cdot 89^{3} + 21\cdot 89^{4} + 54\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,4)(2,3,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
