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