Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 13.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 75\cdot 89 + 18\cdot 89^{2} + 19\cdot 89^{3} + 61\cdot 89^{4} + 2\cdot 89^{5} + 53\cdot 89^{6} + 3\cdot 89^{7} + 71\cdot 89^{8} + 54\cdot 89^{9} + 17\cdot 89^{10} + 62\cdot 89^{11} + 30\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 + 15\cdot 89 + 60\cdot 89^{2} + 3\cdot 89^{3} + 54\cdot 89^{4} + 52\cdot 89^{5} + 45\cdot 89^{6} + 54\cdot 89^{7} + 58\cdot 89^{8} + 71\cdot 89^{9} + 56\cdot 89^{10} + 11\cdot 89^{11} + 32\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 30\cdot 89 + 60\cdot 89^{2} + 49\cdot 89^{3} + 9\cdot 89^{4} + 10\cdot 89^{5} + 63\cdot 89^{6} + 47\cdot 89^{7} + 49\cdot 89^{8} + 52\cdot 89^{9} + 65\cdot 89^{10} + 87\cdot 89^{11} + 63\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 62\cdot 89 + 29\cdot 89^{2} + 51\cdot 89^{3} + 75\cdot 89^{4} + 27\cdot 89^{5} + 66\cdot 89^{6} + 12\cdot 89^{7} + 59\cdot 89^{8} + 63\cdot 89^{9} + 45\cdot 89^{10} + 42\cdot 89^{11} + 44\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 26\cdot 89 + 59\cdot 89^{2} + 37\cdot 89^{3} + 13\cdot 89^{4} + 61\cdot 89^{5} + 22\cdot 89^{6} + 76\cdot 89^{7} + 29\cdot 89^{8} + 25\cdot 89^{9} + 43\cdot 89^{10} + 46\cdot 89^{11} + 44\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 58\cdot 89 + 28\cdot 89^{2} + 39\cdot 89^{3} + 79\cdot 89^{4} + 78\cdot 89^{5} + 25\cdot 89^{6} + 41\cdot 89^{7} + 39\cdot 89^{8} + 36\cdot 89^{9} + 23\cdot 89^{10} + 89^{11} + 25\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 54 + 73\cdot 89 + 28\cdot 89^{2} + 85\cdot 89^{3} + 34\cdot 89^{4} + 36\cdot 89^{5} + 43\cdot 89^{6} + 34\cdot 89^{7} + 30\cdot 89^{8} + 17\cdot 89^{9} + 32\cdot 89^{10} + 77\cdot 89^{11} + 56\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 73 + 13\cdot 89 + 70\cdot 89^{2} + 69\cdot 89^{3} + 27\cdot 89^{4} + 86\cdot 89^{5} + 35\cdot 89^{6} + 85\cdot 89^{7} + 17\cdot 89^{8} + 34\cdot 89^{9} + 71\cdot 89^{10} + 26\cdot 89^{11} + 58\cdot 89^{12} +O\left(89^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4,8,6,7,5)$ |
| $(3,6)(4,5)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $0$ |
| $4$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $0$ |
| $4$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
| $4$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
