Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 9\cdot 13 + 12\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{4} + 7\cdot 13^{5} + 7\cdot 13^{6} + 5\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 13 + 4\cdot 13^{2} + 2\cdot 13^{3} + 5\cdot 13^{4} + 5\cdot 13^{5} + 10\cdot 13^{6} + 11\cdot 13^{7} + 2\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 8\cdot 13 + 8\cdot 13^{3} + 9\cdot 13^{4} + 8\cdot 13^{5} + 2\cdot 13^{7} + 8\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 12\cdot 13 + 10\cdot 13^{2} + 11\cdot 13^{3} + 11\cdot 13^{4} + 5\cdot 13^{5} + 12\cdot 13^{6} + 5\cdot 13^{7} + 2\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 2\cdot 13^{2} + 13^{3} + 13^{4} + 7\cdot 13^{5} + 7\cdot 13^{7} + 10\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 + 4\cdot 13 + 12\cdot 13^{2} + 4\cdot 13^{3} + 3\cdot 13^{4} + 4\cdot 13^{5} + 12\cdot 13^{6} + 10\cdot 13^{7} + 4\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 11 + 11\cdot 13 + 8\cdot 13^{2} + 10\cdot 13^{3} + 7\cdot 13^{4} + 7\cdot 13^{5} + 2\cdot 13^{6} + 13^{7} + 10\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 12 + 3\cdot 13 + 3\cdot 13^{4} + 5\cdot 13^{5} + 5\cdot 13^{6} + 12\cdot 13^{7} + 7\cdot 13^{8} +O\left(13^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
