Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 3\cdot 19 + 3\cdot 19^{2} + 10\cdot 19^{3} + 11\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 14\cdot 19 + 14\cdot 19^{3} + 5\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 4\cdot 19 + 18\cdot 19^{2} + 4\cdot 19^{3} + 13\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 15\cdot 19 + 15\cdot 19^{2} + 8\cdot 19^{3} + 7\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)(2,3)$ |
| $(1,2,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,2,4,3)$ |
$\zeta_{4}$ |
$-\zeta_{4}$ |
| $1$ |
$4$ |
$(1,3,4,2)$ |
$-\zeta_{4}$ |
$\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
