Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 16 + \left(7 a + 11\right)\cdot 19 + \left(14 a + 8\right)\cdot 19^{2} + \left(17 a + 2\right)\cdot 19^{3} + \left(2 a + 4\right)\cdot 19^{4} + \left(5 a + 9\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 11 + \left(11 a + 5\right)\cdot 19 + \left(4 a + 15\right)\cdot 19^{2} + \left(a + 5\right)\cdot 19^{3} + \left(16 a + 8\right)\cdot 19^{4} + \left(13 a + 11\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 9 + \left(9 a + 13\right)\cdot 19 + 2\cdot 19^{2} + \left(8 a + 14\right)\cdot 19^{3} + \left(17 a + 6\right)\cdot 19^{4} + \left(17 a + 13\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 9\cdot 19 + 17\cdot 19^{2} + 12\cdot 19^{3} + 2\cdot 19^{4} + 9\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 15 + \left(9 a + 16\right)\cdot 19 + \left(18 a + 12\right)\cdot 19^{2} + \left(10 a + 2\right)\cdot 19^{3} + \left(a + 16\right)\cdot 19^{4} + \left(a + 13\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,5,4,3,2)$ |
| $(1,3,4,2)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $5$ | $4$ | $(1,5,2,3)$ | $0$ |
| $5$ | $4$ | $(1,3,2,5)$ | $0$ |
| $4$ | $5$ | $(1,5,4,3,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
