Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 5 + \left(17 a + 15\right)\cdot 19 + \left(12 a + 2\right)\cdot 19^{2} + 6\cdot 19^{3} + \left(4 a + 2\right)\cdot 19^{4} + \left(6 a + 14\right)\cdot 19^{5} + \left(9 a + 12\right)\cdot 19^{6} + \left(16 a + 15\right)\cdot 19^{7} + \left(18 a + 3\right)\cdot 19^{8} + \left(6 a + 17\right)\cdot 19^{9} + \left(12 a + 4\right)\cdot 19^{10} + \left(15 a + 16\right)\cdot 19^{11} +O\left(19^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 14\cdot 19 + 4\cdot 19^{2} + 7\cdot 19^{3} + 19^{4} + 10\cdot 19^{5} + 4\cdot 19^{6} + 13\cdot 19^{7} + 12\cdot 19^{8} + 8\cdot 19^{9} + 19^{10} + 9\cdot 19^{11} +O\left(19^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 16 + \left(17 a + 3\right)\cdot 19 + \left(12 a + 2\right)\cdot 19^{2} + 6\cdot 19^{3} + \left(4 a + 13\right)\cdot 19^{4} + \left(6 a + 2\right)\cdot 19^{5} + \left(9 a + 3\right)\cdot 19^{6} + \left(16 a + 15\right)\cdot 19^{7} + \left(18 a + 12\right)\cdot 19^{8} + \left(6 a + 13\right)\cdot 19^{9} + \left(12 a + 8\right)\cdot 19^{10} + \left(15 a + 18\right)\cdot 19^{11} +O\left(19^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 4\cdot 19 + 14\cdot 19^{2} + 11\cdot 19^{3} + 17\cdot 19^{4} + 8\cdot 19^{5} + 14\cdot 19^{6} + 5\cdot 19^{7} + 6\cdot 19^{8} + 10\cdot 19^{9} + 17\cdot 19^{10} + 9\cdot 19^{11} +O\left(19^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 4 + \left(a + 15\right)\cdot 19 + \left(6 a + 16\right)\cdot 19^{2} + \left(18 a + 12\right)\cdot 19^{3} + \left(14 a + 5\right)\cdot 19^{4} + \left(12 a + 16\right)\cdot 19^{5} + \left(9 a + 15\right)\cdot 19^{6} + \left(2 a + 3\right)\cdot 19^{7} + 6\cdot 19^{8} + \left(12 a + 5\right)\cdot 19^{9} + \left(6 a + 10\right)\cdot 19^{10} + 3 a\cdot 19^{11} +O\left(19^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 15 + \left(a + 3\right)\cdot 19 + \left(6 a + 16\right)\cdot 19^{2} + \left(18 a + 12\right)\cdot 19^{3} + \left(14 a + 16\right)\cdot 19^{4} + \left(12 a + 4\right)\cdot 19^{5} + \left(9 a + 6\right)\cdot 19^{6} + \left(2 a + 3\right)\cdot 19^{7} + 15\cdot 19^{8} + \left(12 a + 1\right)\cdot 19^{9} + \left(6 a + 14\right)\cdot 19^{10} + \left(3 a + 2\right)\cdot 19^{11} +O\left(19^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(4,6)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)(2,4)$ |
$-1$ |
| $3$ |
$2$ |
$(2,4)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(4,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-1$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,6,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,6)(2,5,4,3)$ |
$-1$ |
| $8$ |
$6$ |
$(1,3,2,6,5,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
