Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 49\cdot 83 + 18\cdot 83^{2} + 81\cdot 83^{3} + 54\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 19\cdot 83 + 52\cdot 83^{2} + 3\cdot 83^{3} + 37\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 52\cdot 83 + 29\cdot 83^{2} + 48\cdot 83^{3} + 62\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 74 + 44\cdot 83 + 65\cdot 83^{2} + 32\cdot 83^{3} + 11\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,3,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
