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