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