Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 82\cdot 83 + 9\cdot 83^{2} + 30\cdot 83^{3} + 74\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 76\cdot 83 + 33\cdot 83^{2} + 16\cdot 83^{3} + 44\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 6\cdot 83 + 49\cdot 83^{2} + 66\cdot 83^{3} + 38\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 73\cdot 83^{2} + 52\cdot 83^{3} + 8\cdot 83^{4} +O\left(83^{ 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.
