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