Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 58\cdot 89 + 9\cdot 89^{2} + 69\cdot 89^{3} + 53\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 60\cdot 89 + 6\cdot 89^{2} + 64\cdot 89^{3} + 86\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 83\cdot 89 + 87\cdot 89^{2} + 6\cdot 89^{3} + 31\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 + 65\cdot 89 + 73\cdot 89^{2} + 37\cdot 89^{3} + 6\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 + 63\cdot 89 + 50\cdot 89^{2} + 52\cdot 89^{3} + 78\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 + 30\cdot 89 + 70\cdot 89^{2} + 9\cdot 89^{3} + 77\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 26\cdot 89 + 67\cdot 89^{3} + 69\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 76 + 56\cdot 89 + 56\cdot 89^{2} + 48\cdot 89^{3} + 41\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,7,8,6)$ |
| $(5,8)(6,7)$ |
| $(1,5)(2,8)(3,7)(4,6)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,3,2,4)(5,7,8,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,3)(5,6,8,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,2,5)(3,6,4,7)$ | $0$ |
| $2$ | $4$ | $(1,7,2,6)(3,8,4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
