Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 27\cdot 89 + 25\cdot 89^{2} + 49\cdot 89^{3} + 27\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 5\cdot 89 + 88\cdot 89^{2} + 62\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 75\cdot 89 + 7\cdot 89^{2} + 26\cdot 89^{3} + 77\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 68\cdot 89 + 60\cdot 89^{2} + 24\cdot 89^{3} + 74\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 62\cdot 89 + 29\cdot 89^{2} + 10\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 26\cdot 89 + 30\cdot 89^{2} + 56\cdot 89^{3} + 34\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 64\cdot 89 + 25\cdot 89^{2} + 50\cdot 89^{3} + 6\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 84 + 26\cdot 89 + 88\cdot 89^{2} + 75\cdot 89^{3} + 46\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,7,6,3,4,5,8)$ |
| $(1,5,3,7)(2,8,4,6)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $-1$ |
| $1$ | $4$ | $(1,7,3,5)(2,6,4,8)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,5,3,7)(2,8,4,6)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,2,7,6,3,4,5,8)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,6,5,2,3,8,7,4)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,7,8,3,2,5,6)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,8,5,4,3,6,7,2)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
