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