Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7\cdot 13 + 5\cdot 13^{2} + 12\cdot 13^{3} + 6\cdot 13^{4} + 8\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 5\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 11\cdot 13^{4} + 10\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 9\cdot 13 + 2\cdot 13^{2} + 3\cdot 13^{3} + 2\cdot 13^{4} + 3\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 3\cdot 13 + 6\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 3\cdot 13^{5} +O\left(13^{ 6 }\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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
