Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 58\cdot 89 + 57\cdot 89^{2} + 73\cdot 89^{3} + 76\cdot 89^{4} + 68\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 51\cdot 89 + 63\cdot 89^{2} + 55\cdot 89^{3} + 53\cdot 89^{4} + 63\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 56\cdot 89 + 75\cdot 89^{2} + 44\cdot 89^{3} + 38\cdot 89^{4} + 47\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 88\cdot 89 + 85\cdot 89^{2} + 77\cdot 89^{3} + 55\cdot 89^{4} + 64\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 56\cdot 89 + 67\cdot 89^{2} + 80\cdot 89^{3} + 15\cdot 89^{4} + 26\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 67 + 88\cdot 89 + 32\cdot 89^{2} + 69\cdot 89^{3} + 66\cdot 89^{4} + 11\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 68 + 48\cdot 89 + 57\cdot 89^{2} + 60\cdot 89^{3} + 25\cdot 89^{4} + 53\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 73 + 85\cdot 89 + 3\cdot 89^{2} + 71\cdot 89^{3} + 22\cdot 89^{4} + 20\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,7,8)(2,3,6,5)$ |
| $(1,8)(3,5)(4,7)$ |
| $(1,2,8,5,7,6,4,3)$ |
| $(1,7)(2,6)(3,5)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,5)(4,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(3,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,8,7,4)(2,5,6,3)$ | $0$ |
| $2$ | $8$ | $(1,2,8,5,7,6,4,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,4,2,7,3,8,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
