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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 7\cdot 17 + 7\cdot 17^{2} + 17^{3} + 3\cdot 17^{4} + 5\cdot 17^{5} + 13\cdot 17^{6} + 16\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 1 + 6\cdot 17 + \left(a + 12\right)\cdot 17^{2} + 16 a\cdot 17^{3} + \left(10 a + 5\right)\cdot 17^{4} + \left(16 a + 11\right)\cdot 17^{5} + \left(6 a + 16\right)\cdot 17^{6} + \left(6 a + 16\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 9 + \left(5 a + 8\right)\cdot 17 + \left(5 a + 1\right)\cdot 17^{2} + \left(9 a + 6\right)\cdot 17^{3} + \left(a + 9\right)\cdot 17^{4} + \left(13 a + 6\right)\cdot 17^{5} + \left(8 a + 8\right)\cdot 17^{6} + \left(5 a + 10\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 2 + \left(10 a + 6\right)\cdot 17 + \left(15 a + 12\right)\cdot 17^{2} + \left(16 a + 10\right)\cdot 17^{3} + \left(6 a + 1\right)\cdot 17^{4} + \left(6 a + 10\right)\cdot 17^{5} + \left(12 a + 15\right)\cdot 17^{6} + \left(8 a + 12\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 a + 7 + \left(6 a + 11\right)\cdot 17 + a\cdot 17^{2} + 12\cdot 17^{3} + \left(10 a + 8\right)\cdot 17^{4} + \left(10 a + 9\right)\cdot 17^{5} + \left(4 a + 4\right)\cdot 17^{6} + \left(8 a + 9\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 15 + \left(16 a + 9\right)\cdot 17 + \left(15 a + 12\right)\cdot 17^{2} + 15\cdot 17^{3} + \left(6 a + 16\right)\cdot 17^{4} + 16\cdot 17^{5} + \left(10 a + 6\right)\cdot 17^{6} + \left(10 a + 16\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ a + 8 + \left(11 a + 15\right)\cdot 17 + 11 a\cdot 17^{2} + \left(7 a + 10\right)\cdot 17^{3} + \left(15 a + 1\right)\cdot 17^{4} + \left(3 a + 1\right)\cdot 17^{5} + \left(8 a + 4\right)\cdot 17^{6} + \left(11 a + 7\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 12 + 3\cdot 17 + 3\cdot 17^{2} + 11\cdot 17^{3} + 4\cdot 17^{4} + 7\cdot 17^{5} + 15\cdot 17^{6} + 11\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,8,3)(2,7,6,4)$ |
| $(2,4)(3,5)(6,7)$ |
| $(2,7,5)(3,6,4)$ |
| $(1,7,8,4)(2,3,6,5)$ |
| $(1,8)(2,6)(3,5)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,6)(3,5)(4,7)$ |
$-4$ |
| $12$ |
$2$ |
$(2,4)(3,5)(6,7)$ |
$0$ |
| $8$ |
$3$ |
$(1,5,6)(2,8,3)$ |
$1$ |
| $6$ |
$4$ |
$(1,5,8,3)(2,7,6,4)$ |
$0$ |
| $8$ |
$6$ |
$(1,2,5,8,6,3)(4,7)$ |
$-1$ |
| $6$ |
$8$ |
$(1,7,5,6,8,4,3,2)$ |
$0$ |
| $6$ |
$8$ |
$(1,4,5,2,8,7,3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
