Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 5\cdot 89 + 35\cdot 89^{2} + 88\cdot 89^{3} + 6\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 85\cdot 89 + 66\cdot 89^{2} + 70\cdot 89^{3} + 24\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 + 25\cdot 89^{2} + 77\cdot 89^{3} + 50\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 42\cdot 89 + 81\cdot 89^{2} + 82\cdot 89^{3} + 87\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 30\cdot 89 + 56\cdot 89^{2} + 8\cdot 89^{3} + 55\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 + 56\cdot 89 + 38\cdot 89^{2} + 87\cdot 89^{3} + 6\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 78 + 18\cdot 89 + 49\cdot 89^{2} + 30\cdot 89^{3} + 13\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 85 + 26\cdot 89 + 3\cdot 89^{2} + 88\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,8,5,7,2,3,4,6)$ |
| $(1,5,2,4)(3,6,8,7)$ |
| $(3,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,5,2,4)(3,6,8,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,5)(3,7,8,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,2,4)(3,7,8,6)$ | $0$ |
| $2$ | $8$ | $(1,8,5,7,2,3,4,6)$ | $0$ |
| $2$ | $8$ | $(1,7,4,8,2,6,5,3)$ | $0$ |
| $2$ | $8$ | $(1,3,4,7,2,8,5,6)$ | $0$ |
| $2$ | $8$ | $(1,7,5,3,2,6,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
