Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 125 / 1 / b / ii

Table of contents
- Solution ii

Solution

Direct enumeration of the solutions of

y^{2} + y = x^{3} - 4 x + 2

gives

p 357 # {(x, y) \in F_{p}^{2} : y^{2} + y = x^{3} - 4 x + 2} 389 # E (F_{p}) 4910

after adjoining the point at infinity. Therefore

# E (F_{3}) = 4, # E (F_{5}) = 9, # E (F_{7}) = 10.

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