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:
| $r_{ 1 }$ |
$=$ |
$ 5\cdot 19 + 15\cdot 19^{3} + 8\cdot 19^{4} + 19^{5} + 13\cdot 19^{6} + 3\cdot 19^{8} + 17\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a^{2} + 4 a + 14 + \left(6 a^{2} + 13 a\right)\cdot 19 + \left(17 a^{2} + 18 a + 8\right)\cdot 19^{2} + \left(12 a^{2} + 11 a + 17\right)\cdot 19^{3} + \left(14 a^{2} + 6 a + 2\right)\cdot 19^{4} + \left(12 a^{2} + 15 a + 11\right)\cdot 19^{5} + \left(2 a + 10\right)\cdot 19^{6} + \left(16 a^{2} + 6 a + 10\right)\cdot 19^{7} + \left(16 a^{2} + 15 a + 10\right)\cdot 19^{8} + \left(15 a^{2} + 9 a + 1\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a^{2} + 18 a + 7 + \left(15 a^{2} + 9 a + 10\right)\cdot 19 + \left(16 a^{2} + 17 a + 12\right)\cdot 19^{2} + 9 a^{2}19^{3} + \left(14 a^{2} + 15 a + 12\right)\cdot 19^{4} + \left(12 a^{2} + 6\right)\cdot 19^{5} + \left(6 a^{2} + 13 a + 4\right)\cdot 19^{6} + \left(6 a + 15\right)\cdot 19^{7} + \left(2 a^{2} + 13 a + 16\right)\cdot 19^{8} + \left(6 a^{2} + 9 a + 14\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a^{2} + 10 a + 10 + \left(6 a^{2} + 17 a + 5\right)\cdot 19 + \left(8 a^{2} + 9 a + 9\right)\cdot 19^{2} + \left(6 a^{2} + a + 6\right)\cdot 19^{3} + \left(4 a^{2} + 11 a + 13\right)\cdot 19^{4} + \left(7 a^{2} + 11 a + 2\right)\cdot 19^{5} + \left(7 a^{2} + 11 a + 3\right)\cdot 19^{6} + \left(12 a^{2} + a + 7\right)\cdot 19^{7} + \left(18 a^{2} + 2 a + 15\right)\cdot 19^{8} + \left(18 a^{2} + 16 a + 9\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a^{2} + a + 4 + \left(11 a^{2} + 9 a + 13\right)\cdot 19 + \left(17 a + 13\right)\cdot 19^{2} + \left(10 a^{2} + 18 a + 13\right)\cdot 19^{3} + \left(a^{2} + 5 a + 2\right)\cdot 19^{4} + \left(14 a^{2} + 4 a + 10\right)\cdot 19^{5} + \left(a^{2} + 2 a + 16\right)\cdot 19^{6} + \left(15 a^{2} + 18 a + 3\right)\cdot 19^{7} + \left(9 a^{2} + 15 a + 12\right)\cdot 19^{8} + \left(8 a^{2} + 8\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a^{2} + 16 + \left(11 a^{2} + 18\right)\cdot 19 + \left(a^{2} + 3 a + 9\right)\cdot 19^{2} + \left(18 a^{2} + 18 a + 3\right)\cdot 19^{3} + \left(2 a^{2} + 16 a\right)\cdot 19^{4} + \left(11 a^{2} + 13 a + 15\right)\cdot 19^{5} + \left(10 a^{2} + 3 a + 14\right)\cdot 19^{6} + \left(3 a^{2} + 13 a + 4\right)\cdot 19^{7} + \left(7 a^{2} + 8 a + 5\right)\cdot 19^{8} + \left(4 a^{2} + 8 a + 10\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 5 a + 12 + \left(6 a^{2} + 7 a + 17\right)\cdot 19 + \left(12 a^{2} + 9 a\right)\cdot 19^{2} + \left(18 a^{2} + 5 a + 1\right)\cdot 19^{3} + \left(18 a^{2} + a + 8\right)\cdot 19^{4} + \left(17 a^{2} + 11 a + 12\right)\cdot 19^{5} + \left(10 a^{2} + 4 a + 12\right)\cdot 19^{6} + \left(9 a^{2} + 11 a + 18\right)\cdot 19^{7} + \left(2 a^{2} + a + 3\right)\cdot 19^{8} + \left(3 a^{2} + 12 a + 18\right)\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 5 + 8\cdot 19 + 10\cdot 19^{2} + 12\cdot 19^{3} + 11\cdot 19^{4} + 15\cdot 19^{5} + 3\cdot 19^{6} + 5\cdot 19^{7} + 16\cdot 19^{8} + 11\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 10 + 15\cdot 19 + 10\cdot 19^{2} + 5\cdot 19^{3} + 16\cdot 19^{4} + 16\cdot 19^{6} + 9\cdot 19^{7} + 11\cdot 19^{8} + 2\cdot 19^{9} +O\left(19^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,8)(2,7,4)$ |
| $(2,4,7)(3,6,5)$ |
| $(1,2,3)(4,5,9)(6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $1$ |
$3$ |
$(1,8,9)(2,7,4)(3,6,5)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,9,8)(2,4,7)(3,5,6)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $3$ |
$3$ |
$(1,2,3)(4,5,9)(6,8,7)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,3,2)(4,9,5)(6,7,8)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,9,8)(2,7,4)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,8,9)(2,4,7)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,7,6)(2,3,9)(4,5,8)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,6,7)(2,9,3)(4,8,5)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,5,4)(2,8,3)(6,7,9)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,4,5)(2,3,8)(6,9,7)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
