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 values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$5$ |
$(1,4,2,5,3)$ |
$\zeta_{5}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}^{3}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ |
$5$ |
$(1,2,3,4,5)$ |
$\zeta_{5}^{2}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}$ |
$\zeta_{5}^{3}$ |
| $1$ |
$5$ |
$(1,5,4,3,2)$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{2}$ |
| $1$ |
$5$ |
$(1,3,5,2,4)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.
