Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 48\cdot 89 + 85\cdot 89^{2} + 4\cdot 89^{3} + 70\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 31\cdot 89 + 28\cdot 89^{2} + 11\cdot 89^{3} + 52\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 70\cdot 89 + 79\cdot 89^{2} + 59\cdot 89^{3} + 23\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 59\cdot 89 + 73\cdot 89^{2} + 58\cdot 89^{3} + 68\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 27\cdot 89 + 2\cdot 89^{2} + 24\cdot 89^{3} + 39\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 72\cdot 89 + 47\cdot 89^{2} + 60\cdot 89^{3} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 + 2\cdot 89 + 58\cdot 89^{2} + 31\cdot 89^{3} + 30\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 86 + 43\cdot 89 + 69\cdot 89^{2} + 15\cdot 89^{3} + 71\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,4)(3,7)$ |
| $(1,5)(2,4)(3,7)(6,8)$ |
| $(1,6,5,8)(2,7,4,3)$ |
| $(1,7,6,2,5,3,8,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(2,4)(3,7)$ | $0$ |
| $1$ | $4$ | $(1,6,5,8)(2,3,4,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,5,6)(2,7,4,3)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,5,8)(2,7,4,3)$ | $0$ |
| $2$ | $8$ | $(1,7,6,2,5,3,8,4)$ | $0$ |
| $2$ | $8$ | $(1,2,8,7,5,4,6,3)$ | $0$ |
| $2$ | $8$ | $(1,2,6,3,5,4,8,7)$ | $0$ |
| $2$ | $8$ | $(1,3,8,2,5,7,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
