Points using MAGMA

MAGMA calls rational points of a curve $C$ ``places of degree $1$''. MAGMA will find all points in all affine components at the same time. A point in the affine component $C_1$ is denoted $(a:b:1)$. A point in the affine component $C_2$ is denoted $(1:a:b)$. A point in the affine component $C_3$ is denoted $(a:1:b)$.

> Places(C,1);
[
    Place at (0 : 1 : 0),
    Place at (0 : 0 : 1),
    Place at (1 : 0 : 0),
    Place at (4 : 2 : 1),
    Place at (3 : 2 : 1),
    Place at (3 : 4 : 1)
]
> p:=A![3,2];
> p in C;
true (3, 2)

Divisors can be defined as formal sums of points:

> P1:=Places(C,1);
> D:=P1[2]+P1[3];
> Div := DivisorGroup(C);
> D0:=Div!D;
> D0;
Divisor 1*Place at (0 : 0 : 1) + 1*Place at (1 : 0 : 0)
> D:=2*P1[2]+P1[3];
> D0:=Div!D;
> D0;
Divisor 2*Place at (0 : 0 : 1) + 1*Place at (1 : 0 : 0)