Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 4\cdot 19 + 9\cdot 19^{2} + 13\cdot 19^{3} + 15\cdot 19^{4} + 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 18\cdot 19 + 17\cdot 19^{2} + 3\cdot 19^{3} + 3\cdot 19^{4} + 4\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 2\cdot 19 + 2\cdot 19^{2} + 18\cdot 19^{3} + 4\cdot 19^{4} + 10\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 8\cdot 19 + 4\cdot 19^{2} + 11\cdot 19^{3} + 7\cdot 19^{4} + 7\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 5\cdot 19 + 5\cdot 19^{2} + 3\cdot 19^{3} + 16\cdot 19^{4} +O\left(19^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 + 2\cdot 19 + 3\cdot 19^{2} + 4\cdot 19^{3} + 12\cdot 19^{4} + 10\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 + 5\cdot 19 + 16\cdot 19^{2} + 8\cdot 19^{3} + 14\cdot 19^{4} + 4\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 18 + 8\cdot 19 + 17\cdot 19^{2} + 12\cdot 19^{3} + 19^{4} + 17\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,2,8)(3,4,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
