72 2. Elliptic curves (2) Show that if y2 = f(x) is singular, where f(x) ∈ K[x] is a monic cubic polynomial, then the singularity must occur at (δ,0), where δ is a root of f(x). (3) Show that (δ,0) is singular if and only if δ is a double root of f(x). Therefore D = 0 if and only if E is singular. Exercise 2.12.10. Let E/Q : y2 = x3 + 3. Find all the points of E(F7) and verify that N7 satisfies Hasse’s bound. Exercise 2.12.11. Let E/Q : y2 = x3 + Ax + B and let p ≥ 3 be a prime of bad reduction for E/Q. Show that E(Fp) has a unique singular point. Exercise 2.12.12. Prove parts (1) and (3) of Theorem 2.8.5. (Hint: use Definition 2.8.4 and Proposition 2.7.3.) Exercise 2.12.13. Prove Corollary 2.8.6. Exercise 2.12.14. Let E : y2 = x3 − 10081x. Use Sage (or PARI) to find a minimal set of generators for the subgroup that is spanned by all these points on E: (0,0), (−100,90), 10081 100 , 90729 1000 , (−17,408) 907137 6889 , − 559000596 571787 , 1681 16 , 20295 64 , 833 4 , 21063 8 − 161296 1681 , 19960380 68921 , − 6790208 168921 , − 40498852616 69426531 . (Hint: use Theorem 2.7.4 to determine the rank of E/Q.) Exercise 2.12.15. Let E and δ be defined as in Theorem 2.9.3, and suppose P = (x0,y0) is a point on E with y0 = 0. Show: • δ(P ) · δ(O) = δ(P ). • δ((e1, 0)) · δ((e2, 0)) = δ((e1, 0) + (e2,0)). • δ(P ) · δ((e1, 0)) = δ(P + (e1,0)). Exercise 2.12.16. Let E : y2 = x3+Ax+B be an elliptic curve with A, B ∈ Q, and suppose P = (x0,y0) is a point on E, with y0 = 0.

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