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:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 8 + \left(18 a + 1\right)\cdot 19 + \left(a + 3\right)\cdot 19^{2} + \left(10 a + 14\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 12 + 18 a\cdot 19 + \left(a + 12\right)\cdot 19^{2} + \left(10 a + 3\right)\cdot 19^{3} + \left(9 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 5 + 3\cdot 19 + \left(17 a + 12\right)\cdot 19^{2} + \left(8 a + 6\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 14 + \left(18 a + 13\right)\cdot 19 + \left(a + 9\right)\cdot 19^{2} + \left(10 a + 17\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 18 + 9\cdot 19 + \left(17 a + 5\right)\cdot 19^{2} + \left(8 a + 3\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 3 + 9\cdot 19 + \left(17 a + 14\right)\cdot 19^{2} + \left(8 a + 11\right)\cdot 19^{3} + \left(9 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 |
| $(1,5)(2,6)(3,4)$ |
| $(1,2,4)(3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,4,2)(3,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,4,5,2,3)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,3,2,5,4,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
