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