Although I understand the Mumford representation of points on the Jacobian (of a genus 2 hyperelliptic curve), I don't understand how Magma represents such points. I would guess the confusion arises because Magma represents hyperelliptics in a weighted projective space.
Here is an example from the Magma Handbook:
Example CrvHyp_point_creation_jacobian (H125E15)
Points on $y^2 = x^6 - 3x - 1$ and their images on the Jacobian.
> _<x> := PolynomialRing(Rationals());
> C := HyperellipticCurve(x^6-3*x-1);
> J := Jacobian(C);
> ptsC := Points(C : Bound := 100);
> ptsC;
{@ (1 : -1 : 0), (1 : 1 : 0), (-1 : -1 : 3), (-1 : 1 : 3) @}
> ptsJ := [ ptsC[i] - ptsC[1] : i in [2,3,4] ];
> ptsJ;
[ (1, x^3, 2), (x + 1/3, x^3, 2), (x + 1/3, x^3 + 2/27, 2) ]
For example, how does (x + 1/3, x^3 + 2/27, 2) encode (-1:1:3) - (1:-1:0)?
The relevant page in the Magma handbook is:
http://magma.maths.usyd.edu.au/magma/handbook/text/1432#15869
Thanks in advance!