Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 15.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 36\cdot 89 + 77\cdot 89^{2} + 40\cdot 89^{3} + 35\cdot 89^{4} + 48\cdot 89^{5} + 85\cdot 89^{6} + 11\cdot 89^{7} + 80\cdot 89^{8} + 89^{9} + 27\cdot 89^{10} + 63\cdot 89^{11} + 39\cdot 89^{12} + 87\cdot 89^{13} + 64\cdot 89^{14} +O\left(89^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 10\cdot 89 + 53\cdot 89^{2} + 46\cdot 89^{3} + 18\cdot 89^{4} + 22\cdot 89^{5} + 33\cdot 89^{6} + 51\cdot 89^{7} + 45\cdot 89^{8} + 28\cdot 89^{9} + 41\cdot 89^{11} + 25\cdot 89^{12} + 42\cdot 89^{13} + 28\cdot 89^{14} +O\left(89^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 42\cdot 89 + 6\cdot 89^{2} + 57\cdot 89^{3} + 58\cdot 89^{4} + 77\cdot 89^{5} + 11\cdot 89^{6} + 66\cdot 89^{7} + 83\cdot 89^{8} + 24\cdot 89^{9} + 78\cdot 89^{10} + 6\cdot 89^{11} + 19\cdot 89^{12} + 39\cdot 89^{13} + 36\cdot 89^{14} +O\left(89^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 44\cdot 89 + 26\cdot 89^{2} + 38\cdot 89^{3} + 31\cdot 89^{4} + 42\cdot 89^{5} + 9\cdot 89^{6} + 18\cdot 89^{7} + 86\cdot 89^{8} + 72\cdot 89^{9} + 70\cdot 89^{10} + 63\cdot 89^{11} + 47\cdot 89^{12} + 70\cdot 89^{13} + 89^{14} +O\left(89^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 44\cdot 89 + 62\cdot 89^{2} + 50\cdot 89^{3} + 57\cdot 89^{4} + 46\cdot 89^{5} + 79\cdot 89^{6} + 70\cdot 89^{7} + 2\cdot 89^{8} + 16\cdot 89^{9} + 18\cdot 89^{10} + 25\cdot 89^{11} + 41\cdot 89^{12} + 18\cdot 89^{13} + 87\cdot 89^{14} +O\left(89^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 + 46\cdot 89 + 82\cdot 89^{2} + 31\cdot 89^{3} + 30\cdot 89^{4} + 11\cdot 89^{5} + 77\cdot 89^{6} + 22\cdot 89^{7} + 5\cdot 89^{8} + 64\cdot 89^{9} + 10\cdot 89^{10} + 82\cdot 89^{11} + 69\cdot 89^{12} + 49\cdot 89^{13} + 52\cdot 89^{14} +O\left(89^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 67 + 78\cdot 89 + 35\cdot 89^{2} + 42\cdot 89^{3} + 70\cdot 89^{4} + 66\cdot 89^{5} + 55\cdot 89^{6} + 37\cdot 89^{7} + 43\cdot 89^{8} + 60\cdot 89^{9} + 88\cdot 89^{10} + 47\cdot 89^{11} + 63\cdot 89^{12} + 46\cdot 89^{13} + 60\cdot 89^{14} +O\left(89^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 87 + 52\cdot 89 + 11\cdot 89^{2} + 48\cdot 89^{3} + 53\cdot 89^{4} + 40\cdot 89^{5} + 3\cdot 89^{6} + 77\cdot 89^{7} + 8\cdot 89^{8} + 87\cdot 89^{9} + 61\cdot 89^{10} + 25\cdot 89^{11} + 49\cdot 89^{12} + 89^{13} + 24\cdot 89^{14} +O\left(89^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,7)(2,8)(4,5)$ |
| $(1,2,8,7)(3,5,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
