Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 61\cdot 89 + 7\cdot 89^{2} + 9\cdot 89^{3} + 24\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 11\cdot 89 + 66\cdot 89^{2} + 45\cdot 89^{3} + 73\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 32\cdot 89 + 66\cdot 89^{2} + 14\cdot 89^{3} + 70\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 43\cdot 89 + 8\cdot 89^{2} + 6\cdot 89^{3} + 21\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 71\cdot 89 + 35\cdot 89^{2} + 51\cdot 89^{3} + 30\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 2\cdot 89 + 37\cdot 89^{2} + 22\cdot 89^{3} + 13\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 63 + 68\cdot 89 + 58\cdot 89^{2} + 23\cdot 89^{3} + 78\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 82 + 65\cdot 89 + 75\cdot 89^{2} + 4\cdot 89^{3} + 45\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,6)(2,8,3,7)$ |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,4,5,6)(2,8,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
