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