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