Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 60\cdot 89 + 60\cdot 89^{2} + 2\cdot 89^{3} + 5\cdot 89^{4} + 21\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 57\cdot 89 + 82\cdot 89^{2} + 29\cdot 89^{3} + 89^{4} + 37\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 82\cdot 89 + 35\cdot 89^{2} + 54\cdot 89^{3} + 82\cdot 89^{4} + 5\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 + 67\cdot 89 + 87\cdot 89^{2} + 89^{3} + 25\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 21\cdot 89 + 89^{2} + 87\cdot 89^{3} + 88\cdot 89^{4} + 63\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 + 6\cdot 89 + 53\cdot 89^{2} + 34\cdot 89^{3} + 6\cdot 89^{4} + 83\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 72 + 31\cdot 89 + 6\cdot 89^{2} + 59\cdot 89^{3} + 87\cdot 89^{4} + 51\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 87 + 28\cdot 89 + 28\cdot 89^{2} + 86\cdot 89^{3} + 83\cdot 89^{4} + 67\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,6,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
