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