Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 4\cdot 7 + 4\cdot 7^{2} + 5\cdot 7^{3} + 5\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 2\cdot 7 + 2\cdot 7^{2} + 7^{3} + 7^{4} +O\left(7^{ 5 }\right)$ |
Generators of the action on the roots
$ r_{ 1 }, r_{ 2 } $
Character values on conjugacy classes
| Size | Order | Action on
$ r_{ 1 }, r_{ 2 } $
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
