Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 29\cdot 89^{2} + 7\cdot 89^{3} + 46\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 72\cdot 89 + 71\cdot 89^{2} + 47\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 79\cdot 89 + 70\cdot 89^{2} + 39\cdot 89^{3} + 39\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 79\cdot 89 + 14\cdot 89^{2} + 2\cdot 89^{3} + 29\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 83\cdot 89 + 31\cdot 89^{2} + 62\cdot 89^{3} + 3\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 + 23\cdot 89 + 17\cdot 89^{2} + 50\cdot 89^{3} + 77\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 53 + 26\cdot 89 + 55\cdot 89^{2} + 7\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 73 + 80\cdot 89 + 64\cdot 89^{2} + 49\cdot 89^{3} + 6\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,6,8)(2,3,7,5)$ |
| $(1,6)(2,7)(3,5)(4,8)$ |
| $(1,7,6,2)(3,8,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $-2$ |
| $2$ | $4$ | $(1,4,6,8)(2,3,7,5)$ | $0$ |
| $2$ | $4$ | $(1,7,6,2)(3,8,5,4)$ | $0$ |
| $2$ | $4$ | $(1,3,6,5)(2,8,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
