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