Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 84\cdot 89 + 82\cdot 89^{2} + 26\cdot 89^{3} + 32\cdot 89^{4} + 41\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 22\cdot 89 + 22\cdot 89^{2} + 7\cdot 89^{3} + 56\cdot 89^{4} + 31\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 77\cdot 89 + 64\cdot 89^{2} + 10\cdot 89^{3} + 30\cdot 89^{4} + 41\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 11\cdot 89 + 73\cdot 89^{2} + 88\cdot 89^{3} + 61\cdot 89^{4} + 82\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 2\cdot 89 + 43\cdot 89^{2} + 16\cdot 89^{3} + 30\cdot 89^{4} + 35\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 52\cdot 89 + 39\cdot 89^{2} + 65\cdot 89^{3} + 29\cdot 89^{4} + 28\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 + 79\cdot 89 + 67\cdot 89^{2} + 45\cdot 89^{3} + 53\cdot 89^{4} + 18\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 69 + 25\cdot 89 + 51\cdot 89^{2} + 5\cdot 89^{3} + 62\cdot 89^{4} + 76\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,8)(6,7)$ |
| $(1,3)(2,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,5)(2,3,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
