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