Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 + 75\cdot 89 + 39\cdot 89^{2} + 29\cdot 89^{3} + 88\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 + 71\cdot 89 + 69\cdot 89^{2} + 67\cdot 89^{3} + 53\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 14\cdot 89 + 48\cdot 89^{2} + 17\cdot 89^{3} + 78\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 84 + 15\cdot 89 + 20\cdot 89^{2} + 63\cdot 89^{3} + 46\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
