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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