Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 2\cdot 13 + 11\cdot 13^{2} + 12\cdot 13^{3} + 10\cdot 13^{4} + 2\cdot 13^{5} + 9\cdot 13^{6} + 12\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 2\cdot 13 + 5\cdot 13^{2} + 3\cdot 13^{3} + 12\cdot 13^{4} + 9\cdot 13^{5} + 6\cdot 13^{6} + 7\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 10\cdot 13 + 7\cdot 13^{2} + 9\cdot 13^{3} + 3\cdot 13^{5} + 6\cdot 13^{6} + 5\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 + 10\cdot 13 + 13^{2} + 2\cdot 13^{4} + 10\cdot 13^{5} + 3\cdot 13^{6} +O\left(13^{ 8 }\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
