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