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