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How do you know if an isogeny is surjective or not, and how do you tell how many points on E maps to E'? Does the answer lie in the degree of the isogeny function?

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Isogenies are always surjective, but there's a nuance. They are surjective over the algebraic closure. The correct statement would be "for every $\mathbb{F}_{17}$-point on the green curve there are three $\bar{\mathbb{F}}_{17}$-points on the blue curve which map to it".

To find the preimages of a point, write down a polynomial system and find its solutions. A computer algebra system (e.g., SageMath) may help. The number of solutions is obviously related to the degree of the polynomial system.

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  • $\begingroup$ thank you! I have apparently forgot the original definition of algebraic varieties and I’m also thankful for your paper ‘mathematics on isogeny based crypto’ $\endgroup$ – edlothia Aug 23 '18 at 0:39

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