Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 59\cdot 89 + 83\cdot 89^{2} + 7\cdot 89^{3} + 74\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 17\cdot 89^{2} + 41\cdot 89^{3} + 22\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 68\cdot 89 + 66\cdot 89^{2} + 22\cdot 89^{3} + 59\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 36\cdot 89 + 72\cdot 89^{2} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 79\cdot 89 + 36\cdot 89^{2} + 81\cdot 89^{3} + 87\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 52\cdot 89 + 70\cdot 89^{2} + 52\cdot 89^{3} + 54\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 78 + 32\cdot 89 + 11\cdot 89^{2} + 52\cdot 89^{3} + 16\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 80 + 26\cdot 89 + 86\cdot 89^{2} + 7\cdot 89^{3} + 22\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,8)(3,6,4,7)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
| $(1,7,2,6)(3,5,4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $-2$ |
| $2$ | $4$ | $(1,7,2,6)(3,5,4,8)$ | $0$ |
| $2$ | $4$ | $(1,5,2,8)(3,6,4,7)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
