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.)
