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