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.)