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 :
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,

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).
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 .
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