Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 5 + \left(9 a + 16\right)\cdot 19 + \left(18 a + 5\right)\cdot 19^{2} + \left(8 a + 8\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} + \left(11 a + 3\right)\cdot 19^{5} + \left(10 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 19 + 4\cdot 19^{2} + 14\cdot 19^{3} + 5\cdot 19^{4} + 11\cdot 19^{5} + 18\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 9 + \left(12 a + 10\right)\cdot 19 + \left(6 a + 9\right)\cdot 19^{2} + \left(18 a + 2\right)\cdot 19^{3} + \left(7 a + 11\right)\cdot 19^{4} + \left(17 a + 2\right)\cdot 19^{5} + \left(4 a + 11\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 7 + \left(6 a + 6\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + 14\cdot 19^{3} + 11 a\cdot 19^{4} + \left(a + 12\right)\cdot 19^{5} + \left(14 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 8 + \left(9 a + 3\right)\cdot 19 + 15\cdot 19^{2} + \left(10 a + 17\right)\cdot 19^{3} + \left(13 a + 17\right)\cdot 19^{4} + \left(7 a + 8\right)\cdot 19^{5} + \left(8 a + 4\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
