EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 8, No. 1, 2015, 126-134
ISSN 1307-5543 – www.ejpam.com

On the Group of the Elliptic Curve y2 = x3 + 4px

Naser Zamani1,⇤, Arman Shams2

1 Faculty of Mathematical Science, University of Mohaghegh Ardabili, Ardabil, Iran
2 Department of mathematics, Shahid Madani University, Tabriz, Iran

Abstract. In this paper we study the group structure of the elliptic curves E : y2 = x3 + 4px , where p
is 3, 5 or a prime of the form u4 + v4 for positive integers u, v.
2010 Mathematics Subject Classifications: 11G05
Key Words and Phrases: Elliptic curves, Rank, Isogenous, Selmer group

1. Introduction

Let E denote an elliptic curve over Q and � = E(Q) be the set of all rational points on E.
A seminal Theorem of Mordell-Weil asserts that � is a finitely generated Abelian group in a
natural way with zero element O . We put � = T �F where T and F are the torsion and
maximal free subgroups of � respectively. By the rank of E, rank(E), we mean the rank of
F . Hence the rank of E is positive if and only if E possesses an infinity of rational points.
Computational works show that a typical elliptic curve has more small rank [1, 9].

Let p be a prime number and consider the curve E = E4p : y2 = x3 + 4px . We study the
group � and show that T = Z2. By combining some facts of [4], a result on the Selmer group
of � and that of its isogenous �̃ will be given. Next, when p = 3,5, u4+ v4 for positive integers
u, v, some results on the rank of E are presented. Although, it can be find some similar results
concerning the 2-isogenous of E in the literatures ([5, 8]), which imply some of our results,
our method of study completely differs from those.

2. Preliminaries

We begin with the following proposition which shows some properties of �.

Proposition 1. Let Q = (x 0, y 0), P = (x , y) be two points of E such that x 0 2 Z and Q = 2P.
Then x 2 Z is even.

⇤Corresponding author.

Email addresses: zamanin@uma.ac.ir, naserzaka@yahoo.com (N. Zamani),
shzarghar.arman@gmail.com (A. Shams)

http://www.ejpam.com 126 c� 2015 EJPAM All rights reserved.



N. Zamani, A. Shams / Eur. J. Pure Appl. Math, 8 (2015), 126-134 127

Proof. Let x = a/b, gcd(a, b) = 1 and x /2 Z. According to the group law of �, one can see
that

x 0 = (a2 � 4pb2)2

4ab(a2 + 4pb2)
.

Hence, (a2�4pb2)2�4ab(a2+4pb2)x 0 = 0 which gives that a4 ⌘ 0 (mod 4) and so a is even.
Also, we have b | (a2 � 4pb2)2. Thus b is either even or b = ±1. The first case contradicts
gcd(a, b) = 1. Thus we must have b = ±1 and x 2 Z is even.

Lemma 1. For any prime p, the point 0= (0,0) is the only element of order 2 in �.

Proof. Suppose the contrary, 0 6= P = (x , y) 2 � is of order 2. Thus 2P = O and hence
(x , y) = (x ,�y). Then x 6= 0, y = 0 and x3 + 4px = 0. Setting x = a/b and gcd(a, b) = 1,
we get a3 + 4pab2 = 0. Hence b2 | a3. Since a, b are coprime, so b = ±1, i.e. x 2 Z. But, we
have p = x3/(�4x) = x2/(�4)< 0, a contradiction.

Proposition 2. For any prime p, there is no point of order 3 in �.

Proof. On the contrary, we suppose P = (x , y) 2 � is of order 3, i.e. 2P = �P. Let
P = (x , y), 2P = (x 0, y 0). Hence (x 0, y 0) = �(x , y) = (x ,�y), so x 0 = x . On the other hand,
from duplication formula, we have

x = x 0 = (x2 � 4p)2

4(x3 + 4px)
.

Thus, 16p2 � 24x2p� 3x4 = 0 is a quadratic polynomial in variable p. Therefore,

p =
12x2 ±p�0

16
with �0 = 192x4.

Since �0 is not square, then p /2 N, a contradiction.

The following is one of our main results.

Theorem 1. For any prime p, T ⇠= Z2.

Proof. By Lemma 1, {O , 0} ✓ T . Let P := (x , y) 2 T \ {O , 0}. By Lutz-Nagell theorem, x
and y are integers such that y2 divides the discriminant �= 28p3 of the curve E. Thus

y2 = 1,22, 24, 26, 28, p2, 22p2, 24p2, 26p2, 28p2.

We list the computations done with 2P = (x2, y2) where

x2 =
(3x2 + 4p)2

4y2
� 2x =

(x2 � 4p)2

4(x3 + 4px)

in the following table:



N. Zamani, A. Shams / Eur. J. Pure Appl. Math, 8 (2015), 126-134 128

Table 1: Computations with 2P = (x2, y2)

y2 x (x , y2; p) x2
1 ±1 – –
4 ±1,±2,±4 – –
16 ±1,±2,±4,±8,±16 – –
64 ±1,±2,±4,±8,±16,±32,±64 (2, 64; 7) 9

4
256 ±1,±2,±4,±8,±16,±32,±64,

±128,±256 (2, 256;31) 225
16

p2 ±1,±p,±p2 – –
4p2 ±1,±2,±4,±p,±2p,±4p,±p2,

±2p2,±4p2 – –
16p2 ±1,±2,±4,±8,±16,±p,±2p,

±4p,±8p,±16p – –
64p2 ±1,±2,±4,±8,±16,±32,±64,

±p,±2p,±4p,±8p,±16p,±32p
±64p,±p2,±2p2,±4p2,±8p2,
±16p2,±32p2,±64p2,±64p2 (14, 3136; 7) 9

4
256p2 ±1,±2,±4,±8,±16,±32,±64,

±128,±256,±p,±2p,±4p,±8p,
±16p,±32p,±64p,±128p,±256
±p2,±2p2,±4p2,±8p2,±16p2

±32p2,±64p2,±128p2,±256p2 (62, 246016; 31) 225
16

The symbol ‘–’ in Table 1 means that the equation y2 = x3 + 4px has no integer solution
(x , y; p) and hence no solution for x2. We see that x2 is never zero and so 2P can not be of
finite order. This contradicts the fact that 2P 2 T .

3. A Result on Selmer Group of E

In this section, we want to evaluate the Selmer group of E. For ease in access, we recall
some basic facts on the Selmer groups of the elliptic curves [4, 7]. Let E, E0 be elliptic curves
defined over Q and assume that there exists an isogeny ' : E �! E0 over Q with '0 : E0 ! E
its dual. Let K be a field containing Q with Q its integral closure in K. Then there is an exact
sequence

0 �! E['] �! E
'�! E0 �! 0,

of Gal(Q/Q)-modules where E['] = ker('). Taking Galois cohomology, we obtain the exact
sequence

0 �! E0(K)/'(E(K)) �K�! H1(K, E['])
'⇤�! H1(K, E)['] �! 0,



N. Zamani, A. Shams / Eur. J. Pure Appl. Math, 8 (2015), 126-134 129

where H1(K, E)['] is the kernel of '⇤ and �K is the connecting homomorphism. Consider the
following commutative diagram (�q := �Qq

):

0 �! E0(Q)/'(E(Q))
�Q�! H1(Q, E[']) �! H1(Q, E)['] �! 0

# # #
0 �! ⇧ E0(Qq)/'(E(Qq))

⇧�q�! ⇧H1(Qq, E[']) �! ⇧H1(Qq, E)['] �! 0

where the symbol ⇧means the direct product over P1 = {primes}[{1} and q 2 P1. Then,
the '-Selmer group S(')(E/Q) and the Shafarevich-Tate group qq(E/Q) are defined by

S(')(E/Q) = ker{H1(Q, E[']) �! ⇧H1(Qq, E)[']}
and

qq(E/Q) = ker{H1(Q, E) �! ⇧H1(Qq, E)}
respectively. We note that there is another method of calculating the Selmer group. From the
above commutative diagram and the definition of the Selmer group, we have the equivalent
definition

S(')(E/Q) ={x 2 H1(Q, E[']) | resq(x) 2 Im(�q),8q 2 P1}
=

\

q2P1
Im(�q) (1)

where for each q 2 P1, Im(�q) is regarded as the subgroup of the group H1(Q, E[']) and
resq(x) is the residue of x at q.

In the following using some nice results of [4], we are able to calculate the Selmer group
of E.

Theorem 2. Assume that q 2 P1 and let ( , )q be the Hilbert symbol. For a subgroup
V ⇢ Q⇥q /Q⇥2

q we define

V? =
¶

x 2 Q⇥q /Q⇥2

q | (x , y)q = 1, 8y 2 V
©

.

Then we have

(1) Im(�q) = Im(�2) = Im(�02)? = (�4q)

(2) Im(�0q) = (q).

Proof. It follows [4, Theorem 2.1, Propositions 4.1, 4.2].

Corollary 1. Let Ẽ be the simultaneous curve of E. Then, we have S(')(E/Q) = (�4p) and
S('̃)(Ẽ/Q) = (16p).



N. Zamani, A. Shams / Eur. J. Pure Appl. Math, 8 (2015), 126-134 130

Proof. It follows from (1) and the previous theorem that

S(')(E/Q) =Im(�1)\ Im(�2)\ Im(�p)

=(R⇥/R⇥2
)\ (�4p)\ (�4p)

=(�4p)

and

S('
0)(Ẽ/Q) =S('

0)(Ẽ/Q)
=Im(�01)\ Im(�02)\ Im(�0p)
={1}\ (4p)\ (16p)
=(16p).

4. Computation of the Rank of E

In this section we assume that p = u4 + v4 is a prime number with u, v 2 N. We note that

(2(u4 + v4)(u+ v)2, 4(u2 + uv + v2)(u4 + v4)/(u+ v)3)

is a point of E. Let Ẽ be the simultaneous curve of E and �̃ be its corresponding group. We
consider ↵ and ↵̃ be the group homomorphism

↵ :� �! Q⇥/Q⇥2
↵̃ : �̃ �! Q⇥/Q⇥2

↵(P) =

8
><
>:

1 for P = O
�(p) for P = 0
�(x) for x 6= 0

↵̃(P) =

8
><
>:

1 for P = O
�(�p) for P = 0
�(x) for x 6= 0

where P = (x , y) and � is a natural group homomorphism Q⇥ 7�! Q⇥/Q⇥2. To compute the
rank of E we use the well-known formula (see for example [2, 6])

2r =
#↵(�) ·#↵̃(�̃)

4
, r = rank(E). (2)

Here, ↵(�) and ↵̃(�̃) are given as

1,�(p) 2 ↵(�) =�
�(d) : Cd has at least an integral solution for d|4p

 
,

1,�(�p) 2 ↵̃(�̃) =�
�(d̃) : Cd̃ has at least an integral solution for d̃|� 16p

 

where Cd and Cd̃ are Super-Fermat equations [3]:

Cd :d t4 +
4p
d

z4 = w2, t � 1, z � 1, gcd
�
t, 4p/d

�
= 1



N. Zamani, A. Shams / Eur. J. Pure Appl. Math, 8 (2015), 126-134 131

Ced : ed t4 � 16p
ed

z4 = w2, t � 1, z � 1, gcd(t, 16p/d̃) = 1,

with integer solutions (t, z, w). Hence,

d =± 1,±2,±4,±p,±2p,±4p
ed =± 1,±2,±4,±8,±16,±p,±2p,±4p,±8p,±16p,

and so,

↵(�) ✓��(�1),�(±2),�(±p),�(±2p),�(�4p)
 

,

↵̃(�̃) ✓��(�1),�(±2),�(±4),�(±8),�(±16),�(±p),�(±2p),�(±4p),�(16p)
 

,

together with 1,�(p) 2 ↵(�) and 1,�(�16p) 2 ↵̃(�̃). Now, we define

Sd =
�
(t, z, w)|Cd has integer solutions for d 6= 1, 4p

 
,

Sed =
�
(t, z, w)|Ced has integer solutions for d 6= 1,�16p

 
.

According to [2]

9s,9s̃ 2 N such that
X

d|4p

#Sd = 2s � 2,
X

d̃|16p

#Sd̃ = 2s̃ � 2,

where d and d̃ are square free, #Sd = 0 if Sd = ; and #Sd = 1 if Sd 6= ;. Similarly for Sd̃ . By
(2) we conclude that r = s + s̃ � 2. By the closed property of ↵(�) and having a note to the
Table 2, we conclude that

↵(�) = {1,�(2),�(p),�(2p)}.
Also, using Tables 3 and 4, we have

↵̃(�̃) = {1,�(�1),�(p),�(�p)}.
Now, using these two equalities together with (2) gives that r = 2.

Table 2: Elements of Sd

d Cd integer solutions
2 2t4 + 2pz4 = w2 (u± v, 1, 2u2 ± 2uv + 2v2)

2p 2pt4 + 2z4 = w2 (1, u± v, 2u2 ± 2uv + 2v2)

Table 3: Elements of Sd̃ for d̃ > 0

d Cd̃ integer solutions
2 2t4 � 8pz4 = w2 –

2p 2pt4 � 8z4 = w2 –



N. Zamani, A. Shams / Eur. J. Pure Appl. Math, 8 (2015), 126-134 132

Table 4: Elements of Sd̃ for d̃ < 0

d Cd̃ integer solutions
�1 �t4 + 16pz4 = w2 –
�2 �2t4 + 8pz4 = w2 –
�2p �2pt4 + 8z4 = w2 –

In Tables 3 and 4, the symbol ‘–’ shows that the corresponding equation dose not have any
integer solution (t, z, w). One can check this straightforward. For example, concerning C2̃ in
the Table 3, if there is any solution, then we conclude that 2t4 ⌘ 0 (mod 4), a contradiction
with gcd(t,�8p) = 1. Also, concerning C2̃p in the Table 3, if there is any solution (t, z, w),
then we conclude that 2|t which contradicts gcd(t,�8) = 1. Similar arguments can be done
for other cases. The following theorem, thus, has been proved.

Theorem 3. For the elliptic curve E : y2 = x3 + 4px (p = u4 + v4), the Mordell-Weil theorem
holds as following:

�⇠= Z2 �Z2.

As other observations about the rank of E : y2 = x3 + 4px , we also examined rank(E) in
the cases p = 3,5. The resulting illustrations done with MWRANK† have been collected in
Tables 5-7.

Table 5: p = 3

Cd , Cd̃ Legendre value integer solutions
w2 = 2t4 + 6z4

�2
6

�
= �1 Not

w2 = 3t4 + 4z4
�3

4

�
= �1 Not

w2 = 6t4 + 2z4
�6

2

�
= �1 Not

w2 = 2t4 � 24z4
��2

24

�
= �1 Not

w2 = 3t4 � 16z4
��3

16

�
= �1 Not

w2 = 6t4 � 8z4
��6

8

�
= �1 Not

w2 = �t4 + 48z4
��1

48

�
= �1 Not

w2 = �2t4 + 24z4
��2

24

�
= �1 Not

w2 = �3t4 + 16z4
��3

16

�
= �1 Not

w2 = �4t4 + 12z4
��4

12

�
= �1 Not

w2 = �6t4 + 8z4
��6

8

�
= �1 Not

†
http://homepages.warwick.ac.uk/~masgaj/mwrank/



N. Zamani, A. Shams / Eur. J. Pure Appl. Math, 8 (2015), 126-134 133

Table 6: Rank

p rank of E #qq(E/Q)[2]
2 1 1
3 0 1
5 1 1

17 p  106 2 1

Table 7: p = 5

Cd , Cd̃ Legendre value integer solutions
w2 = 2t4 + 10z4

� 2
10

�
= �1 Not

w2 = 4t4 + 5z4
�4

5

�
= 1 (1, 1,3)

w2 = 5t4 + 4z4
�5

4

�
= 1 (1, 1,3)

w2 = 10t4 + 2z4
� 2

10

�
= �1 Not

w2 = 2t4 � 40z4
��2

40

�
= �1 Not

w2 = 5t4 � 16z4
��5

16

�
= �1 Not

w2 = 10t4 � 8z4
��8

10

�
= �1 Not

w2 = �t4 + 80z4
��1

80

�
= �1 Not

w2 = �2t4 + 40z4
��4

20

�
= �1 Not

w2 = �4t4 + 20z4
��4

20

�
= �1 Not

w2 = �5t4 + 16z4
��5

16

�
= �1 Not

w2 = �10t4 + 8z4
� 10�8

�
= �1 Not



REFERENCES 134

References

[1] A. Brumer and O Mc. Guinness. The behaviour of the Mordell-Weil group of elliptic curves,
Bulletin of American Mathematical Society, 23, 375-382, 1990.

[2] J. S. Chahal. Topics in number theory, Kluwer Academic/Plenum Publisher, 1988.

[3] H. Cohen. Number theory: Tools and Diophantine equations, Springer, Vol. I, 2007.

[4] T. Goto. A study on the Selmer groups of elliptic curves with a rational 2-torsion, Kyushu
University, PhD thesis 2002.

[5] T. Kudo and K. Motose. On group structures of some special elliptic curves, Mathematical
Journal of Okayama University, 47, 81-84, 2005.

[6] J. H. Silverman and J. Tate. Rational points on elliptic curves, Springer, 1992.

[7] J. H. Silverman, The arithmetic of elliptic curves, Springer, 2009.

[8] B. K. Spearman. Elliptic curves y2 = x3 � px of rank two, Mathematical Journal of
Okayama University, 49, 183-184, 2007.

[9] D. Zagier and G. Kramarz. Numerical investigations related to the L-series of certain elliptic
curves, Journal of Indian Mathematical Society, 52, 51-69, 1987.