Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 14\cdot 19 + 13\cdot 19^{2} + 14\cdot 19^{3} + 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 10\cdot 19 + 18\cdot 19^{2} + 10\cdot 19^{3} + 16\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 5\cdot 19 + 17\cdot 19^{2} + 4\cdot 19^{3} + 5\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 7\cdot 19^{2} + 17\cdot 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 + 17\cdot 19 + 19^{2} + 15\cdot 19^{3} + 6\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 + 10\cdot 19 + 18\cdot 19^{2} + 18\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 + 16\cdot 19 + 12\cdot 19^{2} + 3\cdot 19^{3} + 6\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 + 19 + 5\cdot 19^{2} + 9\cdot 19^{3} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,4)(3,7)(6,8)$ |
| $(1,3)(2,6)(4,8)(5,7)$ |
| $(1,2,5,4)(3,6,7,8)$ |
| $(1,2,5,4)(3,8,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,7)(6,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,6)(4,8)(5,7)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,3)(4,7)(5,6)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(3,7)(6,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,2,5,4)(3,6,7,8)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,4,5,2)(3,8,7,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,6,5,8)(2,7,4,3)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,4)(3,8,7,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,5,7)(2,6,4,8)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
