2.4. The group structure on E(Q) 31

groups. Since there are no points of infinite order, the rank

of E2/Q is 0, and

E2(Q) = {O, P, 2P, 3P, 4P, 5P } = {O, (2, ±3), (0, ±1), (−1,0)}.

(3) The curve E3/Q :

y2

=

x3

− 2 does not have any rational

torsion points other than O (as we shall see in the next

section). However, the point P = (3,5) is a rational point.

Thus, P must be a point of infinite order and E3(Q) contains

infinitely many distinct rational points. In fact, the rank of

E3 is equal to 1 and P is a generator of all of E3(Q), i.e.,

E3(Q) = {nP : n ∈ Z} and E3(Q)

∼

= Z.

(4) The elliptic curve E4/Q : y2 = x3 + 7105x2 + 1327104x

features both torsion and infinite order points. In fact,

E4(Q)

∼

=

Z/4Z ⊕ Z3. The torsion subgroup is generated

by the point T = (1152,111744) of order 4. The free part is

generated by three points of infinite order:

P1 = (−6912,6912), P2 = (−5832,188568), P3 = (−5400,206280).

Hence

E4(Q) = {aT + bP1 + cP2 + dP3 : a = 0,1,2 or 3 and b, c, d ∈ Z}.

As we mentioned above, the isomorphism E4(Q)

∼

= Z/4Z ⊕

Z3 is not canonical. For instance, E4(Q)

∼

=

T ⊕P1,P2,P3

but also E4(Q)

∼

=

T ⊕ P1,P2,P3 with P1 = P1 + T .

The rank of E/Q is, in a sense, a measurement of the arithmetic

complexity of the elliptic curve. It is not known if there is an upper

bound for the possible values of RE (the largest rank known is 28,

discovered by Noam Elkies; see Andrej Dujella’s website [Duj09] for

up-to-date records and examples of curves with “high” ranks). It has

been conjectured (with some controversy) that ranks can be arbitrar-

ily large; i.e., for all n ∈ N there exists an elliptic curve E over Q

with RE ≥ n. We state this as a conjecture for future reference:

Conjecture 2.4.7 (Conjecture of the rank). Let N ≥ 0 be a natural

number. Then there exists an elliptic curve E defined over Q with

rank RE ≥ N .