Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 35\cdot 89^{2} + 71\cdot 89^{3} + 45\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 22\cdot 89 + 52\cdot 89^{2} + 15\cdot 89^{3} + 35\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 59\cdot 89 + 17\cdot 89^{2} + 75\cdot 89^{3} + 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 53\cdot 89 + 50\cdot 89^{2} + 40\cdot 89^{3} + 76\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 + 35\cdot 89 + 38\cdot 89^{2} + 48\cdot 89^{3} + 12\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 29\cdot 89 + 71\cdot 89^{2} + 13\cdot 89^{3} + 87\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 67 + 66\cdot 89 + 36\cdot 89^{2} + 73\cdot 89^{3} + 53\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 85 + 88\cdot 89 + 53\cdot 89^{2} + 17\cdot 89^{3} + 43\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,2,5)(4,8,6,7)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,5)(4,8,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
