Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 18\cdot 19 + 19^{2} + 4\cdot 19^{3} + 8\cdot 19^{4} + 5\cdot 19^{5} + 14\cdot 19^{6} + 17\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 14\cdot 19 + 2\cdot 19^{2} + 2\cdot 19^{3} + 17\cdot 19^{4} + 9\cdot 19^{5} + 6\cdot 19^{6} +O\left(19^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 9\cdot 19 + 17\cdot 19^{2} + 7\cdot 19^{3} + 17\cdot 19^{4} + 16\cdot 19^{5} + 4\cdot 19^{6} + 5\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 15\cdot 19 + 15\cdot 19^{2} + 4\cdot 19^{3} + 14\cdot 19^{4} + 5\cdot 19^{5} + 12\cdot 19^{6} + 14\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 + 3\cdot 19 + 3\cdot 19^{2} + 14\cdot 19^{3} + 4\cdot 19^{4} + 13\cdot 19^{5} + 6\cdot 19^{6} + 4\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 9\cdot 19 + 19^{2} + 11\cdot 19^{3} + 19^{4} + 2\cdot 19^{5} + 14\cdot 19^{6} + 13\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 15 + 4\cdot 19 + 16\cdot 19^{2} + 16\cdot 19^{3} + 19^{4} + 9\cdot 19^{5} + 12\cdot 19^{6} + 18\cdot 19^{7} +O\left(19^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 18 + 17\cdot 19^{2} + 14\cdot 19^{3} + 10\cdot 19^{4} + 13\cdot 19^{5} + 4\cdot 19^{6} + 19^{7} +O\left(19^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,3)(5,6,8,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
