Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 46\cdot 89 + 58\cdot 89^{2} + 5\cdot 89^{3} + 13\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 88\cdot 89 + 86\cdot 89^{2} + 83\cdot 89^{3} + 81\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 86\cdot 89 + 73\cdot 89^{2} + 54\cdot 89^{3} + 41\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 + 46\cdot 89 + 47\cdot 89^{2} + 33\cdot 89^{3} + 41\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 75\cdot 89 + 83\cdot 89^{2} + 27\cdot 89^{3} + 38\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 86\cdot 89 + 57\cdot 89^{2} + 10\cdot 89^{3} + 34\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 56\cdot 89 + 48\cdot 89^{2} + 32\cdot 89^{3} + 16\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 88 + 47\cdot 89 + 76\cdot 89^{2} + 17\cdot 89^{3} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,8)(3,7)(4,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,8,3,6)(2,5,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
