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