Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 8\cdot 89 + 32\cdot 89^{2} + 36\cdot 89^{3} + 10\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 13\cdot 89 + 22\cdot 89^{2} + 46\cdot 89^{3} + 21\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 2\cdot 89 + 80\cdot 89^{2} + 4\cdot 89^{3} + 52\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 26\cdot 89 + 71\cdot 89^{2} + 39\cdot 89^{3} + 54\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 + 62\cdot 89 + 17\cdot 89^{2} + 49\cdot 89^{3} + 34\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 86\cdot 89 + 8\cdot 89^{2} + 84\cdot 89^{3} + 36\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 + 75\cdot 89 + 66\cdot 89^{2} + 42\cdot 89^{3} + 67\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 72 + 80\cdot 89 + 56\cdot 89^{2} + 52\cdot 89^{3} + 78\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,4,7)(2,8,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,4,7)(2,8,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
