Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 13 + 7\cdot 13^{2} + 5\cdot 13^{3} + 10\cdot 13^{4} + 3\cdot 13^{6} + 6\cdot 13^{7} + 11\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 3\cdot 13 + 2\cdot 13^{2} + 9\cdot 13^{3} + 5\cdot 13^{4} + 4\cdot 13^{5} + 10\cdot 13^{6} + 5\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 9\cdot 13 + 10\cdot 13^{2} + 3\cdot 13^{3} + 7\cdot 13^{4} + 8\cdot 13^{5} + 2\cdot 13^{6} + 12\cdot 13^{7} + 7\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 11\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 2\cdot 13^{4} + 12\cdot 13^{5} + 9\cdot 13^{6} + 6\cdot 13^{7} + 13^{8} +O\left(13^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,3)$ |
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.
