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