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