Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 + 4\cdot 89 + 20\cdot 89^{2} + 33\cdot 89^{3} + 72\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 + 56\cdot 89 + 63\cdot 89^{2} + 61\cdot 89^{3} + 67\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 + 70\cdot 89 + 41\cdot 89^{2} + 39\cdot 89^{3} + 13\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 46\cdot 89 + 52\cdot 89^{2} + 43\cdot 89^{3} + 24\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4,3,2)$ |
| $(1,3)(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
