Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 8 + \left(16 a + 11\right)\cdot 17 + \left(11 a + 4\right)\cdot 17^{2} + 16\cdot 17^{3} + \left(11 a + 6\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 3 + 17 + \left(5 a + 1\right)\cdot 17^{2} + \left(16 a + 9\right)\cdot 17^{3} + \left(5 a + 6\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 14 + 14\cdot 17 + \left(5 a + 8\right)\cdot 17^{2} + \left(16 a + 11\right)\cdot 17^{3} + \left(5 a + 8\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 4 + 15\cdot 17 + \left(5 a + 16\right)\cdot 17^{2} + \left(16 a + 4\right)\cdot 17^{3} + 5 a\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 7 + \left(16 a + 14\right)\cdot 17 + \left(11 a + 5\right)\cdot 17^{2} + 3\cdot 17^{3} + \left(11 a + 13\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 1 + \left(16 a + 11\right)\cdot 17 + \left(11 a + 13\right)\cdot 17^{2} + 5\cdot 17^{3} + \left(11 a + 15\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6,4,5,3)$ |
| $(1,4)(2,5)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,6,5)(2,4,3)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,6)(2,3,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,6,4,5,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,5,4,6,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
