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