2.12. Exercises 75 Exercise 2.12.24. (Elliptic curves with non-trivial rank.) The goal here is a systematic way to find curves of rank at least r ≥ 0 without using tables of elliptic curves: (1) (Easy) Find 3 non-isomorphic elliptic curves over Q with rank ≥ 2. You must prove that the rank is at least 2. (To show linear independence, you may use PARI or Sage to calculate the height matrix). (2) (Fair) Find 3 non-isomorphic elliptic curves over Q with rank ≥ 3. (3) (Medium difficulty) Find 3 non-isomorphic elliptic curves over Q with rank ≥ 6. If so, then you can probably find 3 curves of rank ≥ 8 as well. (4) (Significantly harder) Find 3 non-isomorphic elliptic curves over Q of rank ≥ 10. (5) (You would be famous!) Find an elliptic curve over Q of rank ≥ 29. Exercise 2.12.25. Let E be an elliptic curve and suppose that the images of the points P1,P2,...,Pn ∈ E(Q) in E(Q)/2E(Q) generate the group E(Q)/2E(Q). Let G be the subgroup of E(Q) generated by P1,P2,...,Pn. (1) Prove that the index of G in E(Q) is finite, i.e., the quotient group E(Q)/G is finite. (2) Show that, depending on the choice of generators {Pi} of the quotient E(Q)/2E(Q), the size of E(Q)/G may be ar- bitrarily large. Exercise 2.12.26. Fermat’s last theorem shows that x3 + y3 = z3 has no integer solutions with xyz = 0. Find the first d ≥ 1 such that x3 + y3 = dz3 has infinitely many non-trivial solutions, find a generator for the solutions and write down a few examples. (Hint: Example 2.2.3.)
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