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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 7\cdot 17 + 3\cdot 17^{2} + 10\cdot 17^{3} + 17^{4} + 13\cdot 17^{5} + 7\cdot 17^{6} + 4\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 6 + 5\cdot 17 + \left(14 a + 12\right)\cdot 17^{2} + \left(9 a + 7\right)\cdot 17^{3} + \left(14 a + 15\right)\cdot 17^{4} + \left(10 a + 11\right)\cdot 17^{5} + \left(16 a + 11\right)\cdot 17^{6} + \left(12 a + 2\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 1 + \left(16 a + 11\right)\cdot 17 + \left(2 a + 8\right)\cdot 17^{2} + \left(7 a + 3\right)\cdot 17^{3} + \left(2 a + 3\right)\cdot 17^{4} + \left(6 a + 8\right)\cdot 17^{5} + \left(4 a + 16\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 10 + \left(16 a + 5\right)\cdot 17 + \left(2 a + 11\right)\cdot 17^{2} + \left(6 a + 4\right)\cdot 17^{3} + \left(2 a + 5\right)\cdot 17^{4} + \left(6 a + 13\right)\cdot 17^{5} + \left(16 a + 16\right)\cdot 17^{6} + \left(7 a + 14\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 11 + 6 a\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(a + 12\right)\cdot 17^{3} + \left(8 a + 12\right)\cdot 17^{4} + \left(12 a + 13\right)\cdot 17^{5} + \left(14 a + 5\right)\cdot 17^{6} + \left(13 a + 14\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 2 + \left(10 a + 16\right)\cdot 17 + \left(9 a + 11\right)\cdot 17^{2} + \left(15 a + 6\right)\cdot 17^{3} + \left(8 a + 2\right)\cdot 17^{4} + \left(4 a + 1\right)\cdot 17^{5} + \left(2 a + 8\right)\cdot 17^{6} + \left(3 a + 13\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 + 6\cdot 17 + 11\cdot 17^{2} + 14\cdot 17^{3} + 8\cdot 17^{4} + 6\cdot 17^{5} + 7\cdot 17^{6} + 12\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 11 a + 16 + 15\cdot 17 + \left(14 a + 14\right)\cdot 17^{2} + \left(10 a + 7\right)\cdot 17^{3} + \left(14 a + 1\right)\cdot 17^{4} + 10 a\cdot 17^{5} + 10\cdot 17^{6} + \left(9 a + 6\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,6)(2,7,4)$ |
| $(1,5,7,3)(2,4,6,8)$ |
| $(1,8,7,4)(2,5,6,3)$ |
| $(1,2)(4,8)(6,7)$ |
| $(1,7)(2,6)(3,5)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,6)(3,5)(4,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,2)(4,8)(6,7)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,5,4)(3,8,7)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,5,7,3)(2,4,6,8)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,7)(2,4,5,6,8,3)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,4,2,5,7,8,6,3)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,8,2,3,7,4,6,5)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
