Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 + 31\cdot 89 + 12\cdot 89^{2} + 2\cdot 89^{3} + 7\cdot 89^{4} + 4\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 16\cdot 89 + 32\cdot 89^{2} + 27\cdot 89^{3} + 25\cdot 89^{4} + 8\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 69\cdot 89 + 12\cdot 89^{2} + 69\cdot 89^{3} + 16\cdot 89^{4} + 83\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 43\cdot 89 + 49\cdot 89^{2} + 70\cdot 89^{3} + 78\cdot 89^{4} + 55\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 33\cdot 89 + 14\cdot 89^{2} + 36\cdot 89^{3} + 75\cdot 89^{4} + 34\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 72 + 86\cdot 89 + 83\cdot 89^{2} + 77\cdot 89^{3} + 66\cdot 89^{4} + 20\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 81\cdot 89 + 5\cdot 89^{2} + 67\cdot 89^{3} + 63\cdot 89^{4} + 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 87 + 81\cdot 89 + 55\cdot 89^{2} + 5\cdot 89^{3} + 22\cdot 89^{4} + 58\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5)(2,7,6,8)$ |
| $(1,2)(3,8)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,4,5)(2,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
