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