Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 64\cdot 89 + 68\cdot 89^{2} + 87\cdot 89^{3} + 22\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 9\cdot 89 + 69\cdot 89^{2} + 3\cdot 89^{3} + 31\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 24\cdot 89 + 52\cdot 89^{2} + 69\cdot 89^{3} + 48\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 51\cdot 89 + 35\cdot 89^{2} + 48\cdot 89^{3} + 60\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 61 + 59\cdot 89 + 52\cdot 89^{2} + 69\cdot 89^{3} + 77\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 + 16\cdot 89 + 7\cdot 89^{2} + 30\cdot 89^{3} + 7\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 75 + 56\cdot 89 + 64\cdot 89^{2} + 6\cdot 89^{3} + 83\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 82 + 73\cdot 89 + 5\cdot 89^{2} + 40\cdot 89^{3} + 24\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(6,8)$ |
| $(1,7,6,3,2,5,8,4)$ |
| $(3,4)(5,7)$ |
| $(1,6,2,8)(3,7,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,6,2,8)(3,5,4,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,2,6)(3,7,4,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,2,8)(3,7,4,5)$ | $0$ |
| $2$ | $8$ | $(1,7,6,3,2,5,8,4)$ | $0$ |
| $2$ | $8$ | $(1,3,8,7,2,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,8,3,2,7,6,4)$ | $0$ |
| $2$ | $8$ | $(1,3,6,5,2,4,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
