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:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 11 + \left(10 a + 2\right)\cdot 19 + \left(3 a + 13\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(11 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 6 + \left(7 a + 9\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + \left(16 a + 16\right)\cdot 19^{3} + \left(3 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 2 + 7\cdot 19 + \left(3 a + 8\right)\cdot 19^{2} + \left(18 a + 11\right)\cdot 19^{3} + \left(3 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 8 + \left(8 a + 16\right)\cdot 19 + \left(15 a + 5\right)\cdot 19^{2} + \left(15 a + 9\right)\cdot 19^{3} + \left(7 a + 13\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 a + 13 + \left(11 a + 9\right)\cdot 19 + \left(6 a + 2\right)\cdot 19^{2} + \left(2 a + 2\right)\cdot 19^{3} + \left(15 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 17 + \left(18 a + 11\right)\cdot 19 + \left(15 a + 10\right)\cdot 19^{2} + 7\cdot 19^{3} + \left(15 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,2,4,3,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,5,3,4,2,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
