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