2.12. Exercises 71 (2) Let y2 = x(x + 5)(x + 10) and P = (−9,6). Find nP for n = 1,...,12. Compare the x-coordinates of nP with the list given at the end of Example 1.1.1, and write down the next three numbers that belong in the list. Exercise 2.12.6. Let E/Q be an elliptic curve given by a Weierstrass equation of the form y2 = f(x), where f(x) ∈ Z[x] is a monic cubic polynomial with distinct roots (over C). (1) Show that P = (x, y) ∈ E is a torsion point of exact order 2 if and only if y = 0 and f(x) = 0. (2) Let E(Q)[2] be the subgroup of E(Q) formed by those ra- tional points P ∈ E(Q) such that 2P = O. Show that the size of E(Q)[2] may be 1, 2 or 4. (3) Give examples of three elliptic curves defined over Q where the size of E(Q)[2] is 1, 2 and 4, respectively. Exercise 2.12.7. Let Et : y2+(1−t)xy−ty = x3−tx2 with t ∈ Q and Δt = t5(t2 − 11t − 1) = 0. As we saw in Example 2.5.4 (or Appendix E), every curve Et has a subgroup isomorphic to Z/5Z. Use Sage to find elliptic curves with torsion Z/5Z and rank 0, 1 and 2. Also, try to find an elliptic curve Et with rank r, as high as possible. (Note: the highest rank known — as of 6/1/2009 — for an elliptic curve with Z/5Z torsion is 6, discovered by Dujella and Lecacheux in 2001 see [Duj09].) Exercise 2.12.8. Let p ≥ 2 be a prime and Ep : y2 = x3 + p2. Show that there is no torsion point P ∈ Ep(Q) with y(P ) equal to y = ±1, ±p2, ±3p, ±3p2, or ± 3. Prove that Q = (0,p) is a torsion point of exact order 3. Conclude that {O,Q, 2Q} are the only torsion points on Ep(Q). (Note: for p = 3, the point (−2,1) ∈ E3(Q). Show that it is not a torsion point.) Exercise 2.12.9. Prove Proposition 2.6.8, as follows: (1) First show that if f(x) is a polynomial, f (x) its derivative, and f(δ) = f (δ) = 0, then f(x) has a double root at δ.

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