Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 59\cdot 89 + 69\cdot 89^{2} + 71\cdot 89^{3} + 27\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 75\cdot 89 + 57\cdot 89^{2} + 78\cdot 89^{3} + 11\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 65\cdot 89 + 57\cdot 89^{2} + 13\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 66\cdot 89 + 82\cdot 89^{2} + 11\cdot 89^{3} + 10\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 24\cdot 89 + 11\cdot 89^{2} + 31\cdot 89^{3} + 12\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 67 + 82\cdot 89 + 53\cdot 89^{2} + 29\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 66\cdot 89 + 31\cdot 89^{2} + 46\cdot 89^{3} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 86 + 4\cdot 89 + 80\cdot 89^{2} + 72\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 |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,3,4,2)(5,7,8,6)$ |
| $(1,6,4,7)(2,5,3,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$4$ |
$(1,3,4,2)(5,7,8,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,4,7)(2,5,3,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,4,5)(2,6,3,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
