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