Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 4\cdot 13 + 3\cdot 13^{2} + 3\cdot 13^{3} + 9\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 7\cdot 13 + 8\cdot 13^{2} + 13^{3} + 4\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 6\cdot 13 + 2\cdot 13^{3} + 5\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 7\cdot 13 + 6\cdot 13^{3} + 7\cdot 13^{4} +O\left(13^{ 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.
