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