A search for high rank congruent number elliptic curves (original) (raw)
Related papers
On The Rank Of Congruent Elliptic Curves
2017
In this paper, ppp and qqq are two different odd primes. First, We construct the congruent elliptic curves corresponding to ppp, 2p2p2p, pqpqpq, and 2pq,2pq,2pq, then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.
On the high rank pi/3\pi/3pi/3 and 2pi/32\pi/32pi/3-congruent number elliptic curves
arXiv (Cornell University), 2011
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2)n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For fixed θ this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2)x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases. 1991 Mathematics Subject Classification. 11G05. Key words and phrases. θ-congruent number, elliptic curve, Mordell-Weil rank.
On high rank $ pi/3$ and 2pi/32 pi/32pi/3-congruent number elliptic curves
Rocky Mountain Journal of Mathematics, 2014
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2)n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For fixed θ, this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2)x. We study two special cases: θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases. 2010 AMS Mathematics subject classification. Primary 11G05. Keywords and phrases. θ-congruent number, elliptic curve, Mordell-Weil rank.
On the high rank π/3 and 2π/3- congruent number elliptic curves
Rocky Mountain J. Math., 2014
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2 )x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases.
CONSTRUCTION OF HIGH RANK ELLIPTIC CURVES
Journal of Geometric Analysis, 2021
We list a number of strategies for construction of elliptic curves having high rank with special emphasis on those curves induced by Diophantine triples, in which we have contributed more. These strategies have been developed by many authors. In particular we present a new example of a curve, induced by a Diophantine triple, with torsion Z/2Z × Z/4Z and with rank 9 over Q. This is the present record for this kind of curves.
Mathematics of Computation, 2003
We develop an algorithm for bounding the rank of elliptic curves in the family y 2 = x 3 −B x, all of them with torsion group Z/(2 Z) and modular invariant j = 1728. We use it to look for curves of high rank in this family and present four such curves of rank 13 and 22 of rank 12.
The -congruent numbers elliptic curves via a Fermat-type theorem
2020
A positive integer N is called a θ-congruent number if there is a -triangle (a,b,c) with rational sides for which the angle between a and b is equal to θ and its area is N √(r^2-s^2), where θ∈ (0, π), cos(θ)=s/r, and 0 ≤ |s|<r are coprime integers. It is attributed to Fujiwara <cit.> that N is a -congruent number if and only if the elliptic curve E_N^: y^2=x (x+(r+s)N)(x-(r-s)N) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N≠ 1,2,3,6 is a -congruent number if and only if rank of E_N^() is greater than zero. In this paper, we answer positively to a question concerning the existence of methods to create new rational θ-triangle for a θ-congruent number N from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E_N^θ(ℚ)...
High-rank elliptic curves with given torsion group and some applications
Banach Center Publications, 2023
In this survey paper, we describe several methods for constructing elliptic curves with a given torsion group and high rank over the rationals and quadratic fields. We also discuss potential applications of such curves in the elliptic curve factorization method and their role in the construction of rational Diophantine sextuples.