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:
| $r_{ 1 }$ |
$=$ |
$ 8 + 7\cdot 19 + 19^{2} + 11\cdot 19^{3} + 19^{4} + 9\cdot 19^{5} + 18\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 1 + \left(11 a + 13\right)\cdot 19 + \left(8 a + 13\right)\cdot 19^{2} + \left(3 a + 14\right)\cdot 19^{3} + \left(13 a + 10\right)\cdot 19^{4} + \left(5 a + 14\right)\cdot 19^{5} + \left(5 a + 14\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 15 + \left(11 a + 8\right)\cdot 19 + \left(15 a + 13\right)\cdot 19^{2} + \left(15 a + 10\right)\cdot 19^{3} + \left(4 a + 17\right)\cdot 19^{4} + 10 a\cdot 19^{5} + \left(3 a + 8\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 2 + \left(7 a + 14\right)\cdot 19 + \left(3 a + 17\right)\cdot 19^{2} + \left(3 a + 10\right)\cdot 19^{3} + \left(14 a + 6\right)\cdot 19^{4} + \left(8 a + 6\right)\cdot 19^{5} + \left(15 a + 1\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 12 + \left(7 a + 13\right)\cdot 19 + \left(10 a + 10\right)\cdot 19^{2} + \left(15 a + 9\right)\cdot 19^{3} + \left(5 a + 1\right)\cdot 19^{4} + \left(13 a + 7\right)\cdot 19^{5} + \left(13 a + 14\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,3)(4,5)$ |
| $(1,4,2,5,3)$ |
| $(1,4,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,3)(4,5)$ | $0$ |
| $5$ | $4$ | $(1,4,3,5)$ | $0$ |
| $5$ | $4$ | $(1,5,3,4)$ | $0$ |
| $4$ | $5$ | $(1,4,2,5,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
