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^{3} + 4 x + 17 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 a^{2} + 11 a + 3 + \left(a^{2} + 2 a + 12\right)\cdot 19 + \left(4 a^{2} + 6 a + 18\right)\cdot 19^{2} + \left(10 a^{2} + 17 a + 7\right)\cdot 19^{3} + \left(5 a + 6\right)\cdot 19^{4} + \left(11 a^{2} + 6 a + 7\right)\cdot 19^{5} + \left(7 a^{2} + 6 a + 12\right)\cdot 19^{6} + \left(18 a + 3\right)\cdot 19^{7} + \left(13 a^{2} + 11 a + 14\right)\cdot 19^{8} + \left(12 a^{2} + 14 a + 3\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a^{2} + 7 a + 8 + \left(9 a^{2} + 15 a\right)\cdot 19 + \left(10 a^{2} + a + 4\right)\cdot 19^{2} + \left(14 a^{2} + 18 a + 13\right)\cdot 19^{3} + \left(4 a^{2} + 12 a + 17\right)\cdot 19^{4} + \left(15 a^{2} + 14 a + 5\right)\cdot 19^{5} + \left(13 a^{2} + 2 a + 16\right)\cdot 19^{6} + \left(9 a + 10\right)\cdot 19^{7} + \left(7 a^{2} + 3 a + 4\right)\cdot 19^{8} + \left(18 a^{2} + 5 a + 6\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a^{2} + 2 a + 12 + \left(a^{2} + 18 a + 7\right)\cdot 19 + \left(15 a^{2} + 7 a + 13\right)\cdot 19^{2} + \left(12 a^{2} + 12 a + 13\right)\cdot 19^{3} + \left(a^{2} + 10 a + 9\right)\cdot 19^{4} + \left(7 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(18 a^{2} + 15 a + 4\right)\cdot 19^{6} + \left(13 a^{2} + 5 a + 13\right)\cdot 19^{7} + \left(14 a^{2} + 17 a + 12\right)\cdot 19^{8} + \left(a^{2} + 4 a + 2\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a^{2} + 6 a + 17 + \left(12 a^{2} + 5 a + 12\right)\cdot 19 + \left(8 a^{2} + 6 a + 2\right)\cdot 19^{2} + \left(8 a^{2} + 17 a + 2\right)\cdot 19^{3} + \left(17 a^{2} + 17 a + 1\right)\cdot 19^{4} + \left(17 a^{2} + 7\right)\cdot 19^{5} + \left(4 a^{2} + 9 a + 6\right)\cdot 19^{6} + \left(11 a^{2} + 4 a + 12\right)\cdot 19^{7} + \left(18 a^{2} + 4 a + 16\right)\cdot 19^{8} + \left(8 a + 6\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 a^{2} + a + 6 + \left(7 a^{2} + a + 8\right)\cdot 19 + \left(4 a^{2} + 11 a\right)\cdot 19^{2} + \left(13 a^{2} + 2 a + 16\right)\cdot 19^{3} + \left(13 a^{2} + 9\right)\cdot 19^{4} + \left(11 a^{2} + 17 a + 15\right)\cdot 19^{5} + \left(16 a^{2} + 9 a + 4\right)\cdot 19^{6} + \left(17 a^{2} + 10 a + 12\right)\cdot 19^{7} + \left(17 a^{2} + 3 a + 14\right)\cdot 19^{8} + \left(6 a^{2} + 18 a + 13\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 + 3\cdot 19 + 6\cdot 19^{2} + 7\cdot 19^{3} + 13\cdot 19^{4} + 13\cdot 19^{5} + 8\cdot 19^{6} + 9\cdot 19^{8} + 2\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 + 13\cdot 19^{2} + 16\cdot 19^{3} + 12\cdot 19^{4} + 3\cdot 19^{5} + 16\cdot 19^{6} + 6\cdot 19^{7} + 18\cdot 19^{8} + 17\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 4 a^{2} + 11 a + 2 + \left(5 a^{2} + 14 a + 12\right)\cdot 19 + \left(14 a^{2} + 4 a + 17\right)\cdot 19^{2} + \left(16 a^{2} + 8 a + 17\right)\cdot 19^{3} + \left(18 a^{2} + 9 a + 4\right)\cdot 19^{4} + \left(12 a^{2} + 14 a\right)\cdot 19^{5} + \left(14 a^{2} + 13 a + 7\right)\cdot 19^{6} + \left(12 a^{2} + 8 a + 16\right)\cdot 19^{7} + \left(4 a^{2} + 16 a + 4\right)\cdot 19^{8} + \left(16 a^{2} + 5 a + 3\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,8,6)(3,5,7)$ |
| $(1,7,4,6)(2,5,3,8)$ |
| $(1,5,4,8)(2,6,3,7)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $4$ |
$3$ |
$(1,3,6)(2,7,4)$ |
$-1$ |
| $4$ |
$3$ |
$(1,6,3)(2,4,7)$ |
$-1$ |
| $6$ |
$4$ |
$(1,7,4,6)(2,5,3,8)$ |
$0$ |
| $4$ |
$6$ |
$(1,4)(2,5,6,3,8,7)$ |
$1$ |
| $4$ |
$6$ |
$(1,4)(2,7,8,3,6,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
