Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 18.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 13\cdot 17 + 10\cdot 17^{2} + 9\cdot 17^{3} + 8\cdot 17^{4} + 3\cdot 17^{5} + 13\cdot 17^{6} + 16\cdot 17^{7} + 14\cdot 17^{8} + 6\cdot 17^{9} + 5\cdot 17^{10} + 15\cdot 17^{11} + 8\cdot 17^{12} + 14\cdot 17^{13} + 16\cdot 17^{14} + 13\cdot 17^{15} + 12\cdot 17^{16} + 12\cdot 17^{17} +O\left(17^{ 18 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + 7 + \left(6 a + 5\right)\cdot 17 + \left(a + 4\right)\cdot 17^{2} + \left(10 a + 12\right)\cdot 17^{3} + \left(14 a + 3\right)\cdot 17^{4} + \left(2 a + 4\right)\cdot 17^{5} + \left(7 a + 10\right)\cdot 17^{6} + \left(8 a + 15\right)\cdot 17^{7} + 8 a\cdot 17^{8} + \left(2 a + 12\right)\cdot 17^{9} + \left(15 a + 2\right)\cdot 17^{10} + \left(10 a + 2\right)\cdot 17^{11} + \left(7 a + 16\right)\cdot 17^{12} + \left(15 a + 5\right)\cdot 17^{13} + \left(10 a + 12\right)\cdot 17^{14} + \left(3 a + 10\right)\cdot 17^{15} + \left(16 a + 4\right)\cdot 17^{16} + \left(5 a + 15\right)\cdot 17^{17} +O\left(17^{ 18 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 6 + \left(10 a + 13\right)\cdot 17 + \left(15 a + 15\right)\cdot 17^{2} + \left(6 a + 3\right)\cdot 17^{3} + \left(2 a + 8\right)\cdot 17^{4} + \left(14 a + 9\right)\cdot 17^{5} + \left(9 a + 14\right)\cdot 17^{6} + \left(8 a + 16\right)\cdot 17^{7} + 8 a\cdot 17^{8} + \left(14 a + 6\right)\cdot 17^{9} + \left(a + 15\right)\cdot 17^{10} + \left(6 a + 14\right)\cdot 17^{11} + \left(9 a + 12\right)\cdot 17^{12} + \left(a + 13\right)\cdot 17^{13} + \left(6 a + 7\right)\cdot 17^{14} + \left(13 a + 3\right)\cdot 17^{15} + \left(11 a + 5\right)\cdot 17^{17} +O\left(17^{ 18 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 3\cdot 17 + 6\cdot 17^{2} + 7\cdot 17^{3} + 8\cdot 17^{4} + 13\cdot 17^{5} + 3\cdot 17^{6} + 2\cdot 17^{8} + 10\cdot 17^{9} + 11\cdot 17^{10} + 17^{11} + 8\cdot 17^{12} + 2\cdot 17^{13} + 3\cdot 17^{15} + 4\cdot 17^{16} + 4\cdot 17^{17} +O\left(17^{ 18 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 10 + \left(10 a + 11\right)\cdot 17 + \left(15 a + 12\right)\cdot 17^{2} + \left(6 a + 4\right)\cdot 17^{3} + \left(2 a + 13\right)\cdot 17^{4} + \left(14 a + 12\right)\cdot 17^{5} + \left(9 a + 6\right)\cdot 17^{6} + \left(8 a + 1\right)\cdot 17^{7} + \left(8 a + 16\right)\cdot 17^{8} + \left(14 a + 4\right)\cdot 17^{9} + \left(a + 14\right)\cdot 17^{10} + \left(6 a + 14\right)\cdot 17^{11} + 9 a\cdot 17^{12} + \left(a + 11\right)\cdot 17^{13} + \left(6 a + 4\right)\cdot 17^{14} + \left(13 a + 6\right)\cdot 17^{15} + 12\cdot 17^{16} + \left(11 a + 1\right)\cdot 17^{17} +O\left(17^{ 18 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 11 + \left(6 a + 3\right)\cdot 17 + \left(a + 1\right)\cdot 17^{2} + \left(10 a + 13\right)\cdot 17^{3} + \left(14 a + 8\right)\cdot 17^{4} + \left(2 a + 7\right)\cdot 17^{5} + \left(7 a + 2\right)\cdot 17^{6} + 8 a\cdot 17^{7} + \left(8 a + 16\right)\cdot 17^{8} + \left(2 a + 10\right)\cdot 17^{9} + \left(15 a + 1\right)\cdot 17^{10} + \left(10 a + 2\right)\cdot 17^{11} + \left(7 a + 4\right)\cdot 17^{12} + \left(15 a + 3\right)\cdot 17^{13} + \left(10 a + 9\right)\cdot 17^{14} + \left(3 a + 13\right)\cdot 17^{15} + \left(16 a + 16\right)\cdot 17^{16} + \left(5 a + 11\right)\cdot 17^{17} +O\left(17^{ 18 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(2,5)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,6,5,3)$ | $1$ |
| $8$ | $6$ | $(1,3,2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
