Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 2\cdot 89 + 12\cdot 89^{2} + 84\cdot 89^{3} + 67\cdot 89^{4} + 76\cdot 89^{5} + 40\cdot 89^{6} + 75\cdot 89^{7} + 62\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 20\cdot 89 + 42\cdot 89^{2} + 20\cdot 89^{3} + 63\cdot 89^{4} + 4\cdot 89^{5} + 28\cdot 89^{6} + 73\cdot 89^{7} + 81\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 74\cdot 89 + 66\cdot 89^{2} + 7\cdot 89^{3} + 39\cdot 89^{4} + 58\cdot 89^{5} + 64\cdot 89^{6} + 32\cdot 89^{7} + 18\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 46\cdot 89 + 13\cdot 89^{2} + 7\cdot 89^{3} + 13\cdot 89^{4} + 72\cdot 89^{5} + 17\cdot 89^{6} + 67\cdot 89^{7} + 67\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 + 42\cdot 89 + 75\cdot 89^{2} + 81\cdot 89^{3} + 75\cdot 89^{4} + 16\cdot 89^{5} + 71\cdot 89^{6} + 21\cdot 89^{7} + 21\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 + 14\cdot 89 + 22\cdot 89^{2} + 81\cdot 89^{3} + 49\cdot 89^{4} + 30\cdot 89^{5} + 24\cdot 89^{6} + 56\cdot 89^{7} + 70\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 67 + 68\cdot 89 + 46\cdot 89^{2} + 68\cdot 89^{3} + 25\cdot 89^{4} + 84\cdot 89^{5} + 60\cdot 89^{6} + 15\cdot 89^{7} + 7\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 81 + 86\cdot 89 + 76\cdot 89^{2} + 4\cdot 89^{3} + 21\cdot 89^{4} + 12\cdot 89^{5} + 48\cdot 89^{6} + 13\cdot 89^{7} + 26\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,5,8,4)(3,6)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,8)(4,5)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $4$ | $4$ | $(1,6,4,2)(3,5,7,8)$ | $0$ |
| $4$ | $4$ | $(1,2,4,6)(3,8,7,5)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(2,6,7,3)(4,5)$ | $0$ |
| $4$ | $4$ | $(2,3,7,6)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
