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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