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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 14\cdot 19 + 15\cdot 19^{2} + 8\cdot 19^{3} + 15\cdot 19^{4} + 5\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 6 + \left(4 a + 12\right)\cdot 19 + \left(9 a + 8\right)\cdot 19^{2} + \left(a + 18\right)\cdot 19^{3} + \left(12 a + 5\right)\cdot 19^{4} + \left(7 a + 18\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 8 + \left(4 a + 7\right)\cdot 19 + \left(9 a + 5\right)\cdot 19^{2} + \left(a + 8\right)\cdot 19^{3} + \left(12 a + 2\right)\cdot 19^{4} + \left(7 a + 5\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 4\cdot 19 + 3\cdot 19^{2} + 10\cdot 19^{3} + 3\cdot 19^{4} + 13\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 13 + \left(14 a + 6\right)\cdot 19 + \left(9 a + 10\right)\cdot 19^{2} + 17 a\cdot 19^{3} + \left(6 a + 13\right)\cdot 19^{4} + 11 a\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 11 + \left(14 a + 11\right)\cdot 19 + \left(9 a + 13\right)\cdot 19^{2} + \left(17 a + 10\right)\cdot 19^{3} + \left(6 a + 16\right)\cdot 19^{4} + \left(11 a + 13\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(2,6)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,2,4,6,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
