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