## Publication

## On the Equation Y2=(X+p)(X2+p2)

Stroeker, R. J. & Top, J., 1994, In : Rocky mountain journal of mathematics. 24, 3, p. 1135-1161 27 p.Research output: Contribution to journal › Article › Academic › peer-review

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*Rocky mountain journal of mathematics*,

*24*(3), 1135-1161.

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*Rocky mountain journal of mathematics*, vol. 24, no. 3, pp. 1135-1161.

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**On the Equation Y2=(X+p)(X2+p2).** / Stroeker, R J; Top, J.

Research output: Contribution to journal › Article › Academic › peer-review

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TY - JOUR

T1 - On the Equation Y2=(X+p)(X2+p2)

AU - Stroeker, R J

AU - Top, J

PY - 1994

Y1 - 1994

N2 - In this paper the family of elliptic curves over Q given by the equation y(2) = (x + p)(x(2) + p(2)) is studied. It is shown that for p a prime number = +/-3 mod 8, the only rational solution to the equation given here is the one with y = 0. The same is true for p = 2. Standard conjectures predict that the rank of the group of rational points is odd for all other primes p. A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers p. Moreover, this bound can only be attained for certain special prime numbers p = 1 mod 16. Examples of such rank 3 curves are given. Lastly, for certain primes p = 9 mod 16 nontrivial elements in the Shafarevich group of the elliptic curve are constructed. In the literature one finds similar investigations of elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication.

AB - In this paper the family of elliptic curves over Q given by the equation y(2) = (x + p)(x(2) + p(2)) is studied. It is shown that for p a prime number = +/-3 mod 8, the only rational solution to the equation given here is the one with y = 0. The same is true for p = 2. Standard conjectures predict that the rank of the group of rational points is odd for all other primes p. A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers p. Moreover, this bound can only be attained for certain special prime numbers p = 1 mod 16. Examples of such rank 3 curves are given. Lastly, for certain primes p = 9 mod 16 nontrivial elements in the Shafarevich group of the elliptic curve are constructed. In the literature one finds similar investigations of elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication.

KW - ELLIPTIC-CURVES

KW - POINTS

KW - HEIGHT

M3 - Article

VL - 24

SP - 1135

EP - 1161

JO - Rocky mountain journal of mathematics

JF - Rocky mountain journal of mathematics

SN - 0035-7596

IS - 3

ER -

ID: 6405853