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