Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 19\cdot 31 + 23\cdot 31^{2} + 29\cdot 31^{3} + 20\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 16\cdot 31 + 21\cdot 31^{2} + 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 26\cdot 31 + 16\cdot 31^{2} + 30\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
