A family of elliptic curves of rank ≥ 4

Abstract

In this paper we consider a family of elliptic curves of the form y 2 = x 3 −c 2 x +a 2 , where (a, b, c) is a primitive Pythagorean triple. First we show that the rank is positive. Then we construct a subfamily with rank ≥ 4.

Some new families of positive-rank elliptic curves arising from Pythagorean triples

“…Letting −(2t) 2 x + (t 2 − 1) 2 = 0, in (12), yields x = ( t 2 −1 2t ) 2 and y = ( t 2 −1 2t ) 3 . Then, the third point is lying on…”

“…In [3], a subfamily of the elliptic curve y 2 = x 3 − c 2 x + a 2 , with the rank at least 4, has been introduced. In [4], it is proved that the rank of the elliptic curve y 2 = x(x − a 2 )(x − b 2 ) is positive and also in [6] a subfamily of this elliptic curve with the rank at least 2 is obtained.…”

A family of elliptic curves of rank ≥ 5 over Q(m)

“…The rank search of (1) started from this work of Brown and Myers, who gave an infinitude of curves of rank ≥ 3 over Q(m). See, for example, [7,1,19,20,10,8,9], given in the chronological order of discovery. Specifically, in [7], Eikenberg generalized the above result of Brown and Myers and showed there exist infinitely many values of s ∈ Q such that E(1, s)(Q) has rank at least five.…”

“…Then, in [20], she found families of E(1, s) of rank ≥ 3 and ≥ 4 over fields of rational functions in four variables and a family of E(1, s) of rank ≥ 5 parameterized by an elliptic curve of positive rank. In [10], Izadi and Nabardi considered E(r, s) with the caveat that r 2 − s 2 is a square and showed the existence of a family with rank ≥ 4 over Q(m). Recently, in a similar work [9], Izadi and Baghalaghdam have shown that the curve E(t) :…”