Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 15 + 9 a\cdot 19 + \left(17 a + 10\right)\cdot 19^{2} + \left(12 a + 15\right)\cdot 19^{3} + \left(8 a + 5\right)\cdot 19^{4} + \left(2 a + 11\right)\cdot 19^{5} + \left(16 a + 3\right)\cdot 19^{6} + \left(16 a + 9\right)\cdot 19^{7} + \left(14 a + 15\right)\cdot 19^{8} + \left(5 a + 11\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 4 + \left(9 a + 2\right)\cdot 19 + \left(a + 18\right)\cdot 19^{2} + \left(6 a + 10\right)\cdot 19^{3} + \left(10 a + 1\right)\cdot 19^{4} + \left(16 a + 5\right)\cdot 19^{5} + \left(2 a + 17\right)\cdot 19^{6} + \left(2 a + 9\right)\cdot 19^{7} + \left(4 a + 13\right)\cdot 19^{8} + \left(13 a + 2\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 5\cdot 19 + 11\cdot 19^{2} + \left(13 a + 8\right)\cdot 19^{3} + 9 a\cdot 19^{4} + \left(6 a + 2\right)\cdot 19^{5} + \left(6 a + 13\right)\cdot 19^{6} + \left(17 a + 1\right)\cdot 19^{7} + \left(13 a + 12\right)\cdot 19^{8} + \left(8 a + 14\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 3 + \left(6 a + 17\right)\cdot 19 + \left(15 a + 17\right)\cdot 19^{2} + \left(15 a + 18\right)\cdot 19^{3} + \left(16 a + 15\right)\cdot 19^{4} + 8\cdot 19^{5} + \left(12 a + 18\right)\cdot 19^{6} + \left(9 a + 12\right)\cdot 19^{7} + \left(14 a + 10\right)\cdot 19^{8} + 6\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 11 + \left(18 a + 13\right)\cdot 19 + \left(18 a + 10\right)\cdot 19^{2} + \left(5 a + 2\right)\cdot 19^{3} + \left(9 a + 16\right)\cdot 19^{4} + \left(12 a + 17\right)\cdot 19^{5} + \left(12 a + 12\right)\cdot 19^{6} + \left(a + 12\right)\cdot 19^{7} + \left(5 a + 8\right)\cdot 19^{8} + \left(10 a + 9\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 8 + \left(12 a + 18\right)\cdot 19 + \left(3 a + 7\right)\cdot 19^{2} + 3 a\cdot 19^{3} + \left(2 a + 17\right)\cdot 19^{4} + \left(18 a + 11\right)\cdot 19^{5} + \left(6 a + 10\right)\cdot 19^{6} + \left(9 a + 10\right)\cdot 19^{7} + \left(4 a + 15\right)\cdot 19^{8} + \left(18 a + 11\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,6,3,2,4,5)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $3$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
$0$ |
| $1$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$2 \zeta_{3}$ |
$-2 \zeta_{3} - 2$ |
| $1$ |
$3$ |
$(1,4,3)(2,6,5)$ |
$-2 \zeta_{3} - 2$ |
$2 \zeta_{3}$ |
| $2$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$-1$ |
$-1$ |
| $2$ |
$3$ |
$(1,3,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $2$ |
$3$ |
$(1,4,3)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $3$ |
$6$ |
$(1,6,3,2,4,5)$ |
$0$ |
$0$ |
| $3$ |
$6$ |
$(1,5,4,2,3,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
