Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 24\cdot 89 + 49\cdot 89^{2} + 11\cdot 89^{3} + 44\cdot 89^{4} + 30\cdot 89^{6} + 5\cdot 89^{7} + 89^{8} + 49\cdot 89^{9} + 77\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 30\cdot 89 + 74\cdot 89^{2} + 55\cdot 89^{3} + 20\cdot 89^{4} + 71\cdot 89^{5} + 42\cdot 89^{6} + 14\cdot 89^{7} + 23\cdot 89^{8} + 70\cdot 89^{9} + 6\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 42\cdot 89 + 46\cdot 89^{2} + 79\cdot 89^{3} + 35\cdot 89^{4} + 54\cdot 89^{5} + 22\cdot 89^{6} + 32\cdot 89^{7} + 60\cdot 89^{8} + 2\cdot 89^{9} + 61\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 7\cdot 89 + 65\cdot 89^{2} + 54\cdot 89^{3} + 31\cdot 89^{4} + 22\cdot 89^{5} + 89^{6} + 56\cdot 89^{7} + 30\cdot 89^{8} + 49\cdot 89^{9} + 69\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 + 81\cdot 89 + 23\cdot 89^{2} + 34\cdot 89^{3} + 57\cdot 89^{4} + 66\cdot 89^{5} + 87\cdot 89^{6} + 32\cdot 89^{7} + 58\cdot 89^{8} + 39\cdot 89^{9} + 19\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 46\cdot 89 + 42\cdot 89^{2} + 9\cdot 89^{3} + 53\cdot 89^{4} + 34\cdot 89^{5} + 66\cdot 89^{6} + 56\cdot 89^{7} + 28\cdot 89^{8} + 86\cdot 89^{9} + 27\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 77 + 58\cdot 89 + 14\cdot 89^{2} + 33\cdot 89^{3} + 68\cdot 89^{4} + 17\cdot 89^{5} + 46\cdot 89^{6} + 74\cdot 89^{7} + 65\cdot 89^{8} + 18\cdot 89^{9} + 82\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 88 + 64\cdot 89 + 39\cdot 89^{2} + 77\cdot 89^{3} + 44\cdot 89^{4} + 88\cdot 89^{5} + 58\cdot 89^{6} + 83\cdot 89^{7} + 87\cdot 89^{8} + 39\cdot 89^{9} + 11\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6)(2,5)(3,8)(4,7)$ |
| $(1,8)(4,5)$ |
| $(1,7,5,3,8,2,4,6)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,7)(5,8)$ | $0$ |
| $4$ | $2$ | $(2,6)(3,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $0$ |
| $4$ | $8$ | $(1,3,5,2,8,6,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
