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