Families of elliptic curves with rational torsion points of even order (original) (raw)

The divisibility by 2 of rational points on elliptic curves

arXiv: Number Theory, 2017

We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by 2n2^n2n formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields mathbbFq\mathbb{F}_qmathbbFq with a prescribed (small) group E(mathbbFq)E(\mathbb{F}_q)E(mathbbFq). In the last two sections we study 3- and 5-torsion. This paper supercedes arXiv:1605.09279 [math.NT] .

Integral points on a class of elliptic curve

Wuhan University Journal of Natural Sciences, 2006

We prove all integral points of the elliptic curve S =~':~--30.r+133 are ( . r , y ) --( --7 , 0 ) , ( --3 , • • (5 143 3 2 6 , ! 1 1 664 498 677), by using the method of algebraic number theory and p-adic analysis. Furthermore, we develop a computation method to find all integral points on a class of elliptic curve yZ --( . r + a ) ( . r e --a.r --b ) , a , h~ Z , a ~ <41) and find all integer solutions of hyperel liptic Diopbantine equation D S --A.r I --B-r e + (', B z <4AC.

FAMILY OF ELLIPTIC CURVES E(p,q)‎: ‎y2=x2-p2x+q2

Facta Universitatis, Series: Mathematics and Informatics, 2019

In this paper we show that for any two primes p and q greater than 5, theelliptic curve E(p,q) : y2 = x3 − p2x + q2 has rank at least 2. We will also provide twoindependent points on E(p,q). Then we will show that, conjecturally, the family {E(p,q)}contains an infinite subfamily of rank three elliptic curves.

On Prime-Order Elliptic Curves With Embedding Degrees K= 3, 4, and 6

… of the 8th international conference on …, 2008

For an elliptic curve E defined over a finite field IFq, let #E(IFq) = n = hr be the number of IFq-rational points on E, where r is the largest prime divisor of n, and gcd(r, q) = 1. The set of all points of order r in E( ¯IFq) forms a subgroup of E(IFq) denoted by E[r]. For such an ...

Division by 222 of rational points on elliptic curves

St. Petersburg Mathematical Journal, 2018

We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by 2 n formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields Fq with a prescribed (small) group E(Fq). In the last two sections we study 3and 5-torsion.

Elliptic Curves and Biquadrates

2012

−NxThe rank of this family over Q(m,n)is at least 2.Euler constructed a parametric family of integers N expressible in twodifferent ways as a sum of two biquadrates. We prove that for those N thecorresponding family of elliptic curves has rank at least 4 over Q(u). This isan improvement on previous results of Izadi, Khoshnam and Nabardi.