Plotting hyperelliptic curves (and the group law)

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I am trying to create plots of hyperelliptic curves (over the rationals) using SAGE. Moreover, I want to plot the integral points on these curves (within a certain range, obviously) and illustrate the group law on the Jacobian. In other words, I want to create an image similar to the one used here: Hyperelliptic curve plot. Plotting the curve itself is not hard. One may use, say, the following code:

Qx.<x> = QQ[]
f = x^5 + 3
C = HyperellipticCurve(f)
image = plot(C)
image.save('image.png')

I am mainly interested in how to plot integral points on said curve and how to illustrate "addition" of these points (in the sense of addition of the corresponding elements on the Jacobian). Do you know how to do this? Any help would be greatly appreciated.