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