2.3. Integral points 23

is an invertible rational map, defined over Q, that sends (0,1) to

O, and ψ(0, −1) = (0,0). See Exercise 2.12.4. More generally, any

quartic

C : V

2

= aU

4

+ bU

3

+ cU

2

+ dU +

q2

for some a, b, c, d, q ∈ Z is isomorphic over Q to a curve of the form

E :

y2

+ a1xy + a3y =

x3

+

a2x2

+ a4x + a6, also defined over Q. The

isomorphism is given in [Was08], Theorem 2.17, p. 37.

Let E be an elliptic curve over Q given by a Weierstrass equation

E :

y2

+ a1xy + a3y =

x3

+

a2x2

+ a4x + a6, ai ∈ Q.

With a change of variables (x, y) →

(u−2x, u−3y),

we can find the

equation of an elliptic curve isomorphic to E given by

y2

+ (a1u)xy +

(a3u3)y

=

x3

+

(a2u2)x2

+

(a4u4)x

+

(a6u6)

with coeﬃcients aiui ∈ Z for i = 1,2,3,4,6. By the way, this is one

of the reasons for the peculiar numbering of the coeﬃcients ai.

Example 2.2.6. Let E be given by

y2

=

x3

+

x

2

+

5

3

. We may

change variables by x =

X

62

and y =

Y

63

to obtain a new equation

Y

2

=

X3

+ 648X + 77760 with integral coeﬃcients.

2.3. Integral points

In 1929, Siegel proved the following result about integral points E(Z),

i.e., about those points on E with integer coordinates:

Theorem 2.3.1 (Siegel’s theorem; [Sil86], Ch. IX, Thm. 3.1). Let

E/Q be an elliptic curve given by

y2

=

x3

+ Ax + B, with A, B ∈ Z.

Then E has only a finite number of integral points.

Siegel’s theorem is a consequence of a well-known theorem of Roth

on Diophantine approximation. Unfortunately, Siegel’s theorem is not

effective and provides neither a method to find the integral points on

E nor a bound on the number of integral points. However, in [Bak90],

Alan Baker found an alternative proof that provides an explicit upper

bound on the size of the coeﬃcients of an integral solution. More

concretely, if x, y ∈ Z satisfy

y2

=

x3

+ Ax + B, then

max(|x|, |y|)

exp((106

· max(|A|,

|B|))106

).