Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 13\cdot 89 + 59\cdot 89^{2} + 56\cdot 89^{3} + 87\cdot 89^{4} + 32\cdot 89^{5} + 38\cdot 89^{6} + 35\cdot 89^{7} + 35\cdot 89^{8} + 69\cdot 89^{9} + 11\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 5\cdot 89 + 50\cdot 89^{2} + 53\cdot 89^{3} + 72\cdot 89^{4} + 5\cdot 89^{5} + 60\cdot 89^{6} + 85\cdot 89^{7} + 63\cdot 89^{8} + 9\cdot 89^{9} + 37\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 40\cdot 89 + 32\cdot 89^{2} + 16\cdot 89^{3} + 72\cdot 89^{4} + 77\cdot 89^{5} + 31\cdot 89^{6} + 6\cdot 89^{7} + 19\cdot 89^{8} + 35\cdot 89^{9} + 3\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 12\cdot 89 + 45\cdot 89^{2} + 19\cdot 89^{3} + 33\cdot 89^{4} + 77\cdot 89^{5} + 27\cdot 89^{6} + 5\cdot 89^{7} + 53\cdot 89^{8} + 71\cdot 89^{9} + 19\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 76\cdot 89 + 43\cdot 89^{2} + 69\cdot 89^{3} + 55\cdot 89^{4} + 11\cdot 89^{5} + 61\cdot 89^{6} + 83\cdot 89^{7} + 35\cdot 89^{8} + 17\cdot 89^{9} + 69\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 48\cdot 89 + 56\cdot 89^{2} + 72\cdot 89^{3} + 16\cdot 89^{4} + 11\cdot 89^{5} + 57\cdot 89^{6} + 82\cdot 89^{7} + 69\cdot 89^{8} + 53\cdot 89^{9} + 85\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 85 + 83\cdot 89 + 38\cdot 89^{2} + 35\cdot 89^{3} + 16\cdot 89^{4} + 83\cdot 89^{5} + 28\cdot 89^{6} + 3\cdot 89^{7} + 25\cdot 89^{8} + 79\cdot 89^{9} + 51\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 88 + 75\cdot 89 + 29\cdot 89^{2} + 32\cdot 89^{3} + 89^{4} + 56\cdot 89^{5} + 50\cdot 89^{6} + 53\cdot 89^{7} + 53\cdot 89^{8} + 19\cdot 89^{9} + 77\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,3)(4,5)(6,8)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,3)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $4$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
