Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 42\cdot 89 + 78\cdot 89^{2} + 66\cdot 89^{3} + 72\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 82\cdot 89 + 14\cdot 89^{2} + 50\cdot 89^{3} + 22\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 69\cdot 89 + 22\cdot 89^{2} + 66\cdot 89^{3} + 34\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 58\cdot 89 + 53\cdot 89^{2} + 54\cdot 89^{3} + 8\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 + 7\cdot 89 + 12\cdot 89^{2} + 44\cdot 89^{3} + 73\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 + 85\cdot 89 + 42\cdot 89^{2} + 32\cdot 89^{3} + 47\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 + 29\cdot 89 + 8\cdot 89^{2} + 29\cdot 89^{3} + 73\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 82 + 69\cdot 89 + 33\cdot 89^{2} + 12\cdot 89^{3} + 23\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,2,6)(3,8,5,7)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,2,6)(3,8,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
