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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 6 + \left(3 a + 7\right)\cdot 17 + \left(12 a + 10\right)\cdot 17^{2} + \left(8 a + 4\right)\cdot 17^{3} + \left(11 a + 7\right)\cdot 17^{4} + \left(11 a + 4\right)\cdot 17^{5} + \left(2 a + 4\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 13 + \left(8 a + 9\right)\cdot 17 + \left(8 a + 16\right)\cdot 17^{2} + 9 a\cdot 17^{3} + \left(12 a + 10\right)\cdot 17^{4} + \left(9 a + 1\right)\cdot 17^{5} +O\left(17^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 13 + \left(13 a + 3\right)\cdot 17 + \left(4 a + 2\right)\cdot 17^{2} + \left(8 a + 1\right)\cdot 17^{3} + \left(5 a + 10\right)\cdot 17^{4} + \left(5 a + 4\right)\cdot 17^{5} + \left(14 a + 12\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 16\cdot 17 + 2\cdot 17^{2} + 13\cdot 17^{3} + 12\cdot 17^{4} + 5\cdot 17^{5} + 8\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 3 + \left(8 a + 11\right)\cdot 17 + \left(8 a + 16\right)\cdot 17^{2} + \left(7 a + 1\right)\cdot 17^{3} + \left(4 a + 13\right)\cdot 17^{4} + \left(7 a + 15\right)\cdot 17^{5} + \left(16 a + 7\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 + 2\cdot 17 + 2\cdot 17^{2} + 12\cdot 17^{3} + 14\cdot 17^{4} + 17^{5} + 17^{6} +O\left(17^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(3,6)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(3,4)(5,6)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,4,2)(3,6,5)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,4,5,3)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,4,3,2,5)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
