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 + 3 + \left(12 a + 14\right)\cdot 17 + \left(13 a + 15\right)\cdot 17^{2} + 2 a\cdot 17^{3} + \left(4 a + 5\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 5 + \left(12 a + 8\right)\cdot 17 + \left(13 a + 3\right)\cdot 17^{2} + 10 a\cdot 17^{3} + 16\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 16 + \left(4 a + 13\right)\cdot 17 + \left(3 a + 16\right)\cdot 17^{2} + \left(14 a + 6\right)\cdot 17^{3} + \left(12 a + 6\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 15 + \left(4 a + 10\right)\cdot 17 + \left(3 a + 4\right)\cdot 17^{2} + \left(6 a + 14\right)\cdot 17^{3} + \left(16 a + 5\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 + 6\cdot 17 + 13\cdot 17^{2} + 5\cdot 17^{3} + 2\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 + 14\cdot 17 + 13\cdot 17^{2} + 5\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,2,3,4,6)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $15$ | $2$ | $(2,3)(4,6)$ | $0$ |
| $20$ | $3$ | $(1,2,4)(3,6,5)$ | $1$ |
| $30$ | $4$ | $(2,6,3,4)$ | $0$ |
| $24$ | $5$ | $(1,3,5,4,6)$ | $-1$ |
| $20$ | $6$ | $(1,5,2,3,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
