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