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:
| $r_{ 1 }$ |
$=$ |
$ a + 17 + \left(2 a + 4\right)\cdot 19 + \left(15 a + 8\right)\cdot 19^{2} + \left(4 a + 17\right)\cdot 19^{3} + 18\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 18 + \left(16 a + 5\right)\cdot 19 + \left(3 a + 2\right)\cdot 19^{2} + \left(14 a + 7\right)\cdot 19^{3} + \left(18 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 13 + \left(13 a + 16\right)\cdot 19 + \left(5 a + 11\right)\cdot 19^{2} + \left(a + 14\right)\cdot 19^{3} + \left(10 a + 8\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + \left(14 a + 1\right)\cdot 19 + \left(18 a + 13\right)\cdot 19^{2} + \left(2 a + 11\right)\cdot 19^{3} + \left(15 a + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 13 + \left(4 a + 2\right)\cdot 19 + 17\cdot 19^{2} + \left(16 a + 14\right)\cdot 19^{3} + \left(3 a + 4\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a + 17 + \left(5 a + 6\right)\cdot 19 + \left(13 a + 4\right)\cdot 19^{2} + \left(17 a + 10\right)\cdot 19^{3} + \left(8 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)$ |
$3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
