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