Related papers: Biquadrates and Elliptic Curves

Let $P$ be an arbitrary point on an elliptic curve over the complex numbers of the form $y^2=x^3+a_4\,x+a_6$ or of the form $y^2=x^3+a_2\,x^2+a_4\,x$. We provide explicit formulae to compute the points $P/2$, i.e., the points $Q$ such that…

For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*} E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ as a function of $t$ and assuming some standard…

In this paper, we construct a family of elliptic curves with rank $\geq 5$. To do this, we use the Heron formula for a triple $(A^2, B^2, C^2)$ which are not necessarily the three sides of a triangle. It turns out that as parameters of a…

The aim of this paper is to introduce a new family of elliptic curves in the form of $y^2=x(x-a^2)(x-b^2)$ that have positive ranks. We first generate a list of pythagorean triples $(a,b,c)$ and then construct this family of elliptic…

We prove that, when all elliptic curves over $\mathbb{Q}$ are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are…

Starting from the elliptic curve $E: y^2 = x^3 - x$ over $\mathbb{F}_9$, a curve $\mathcal{X}$ over $\mathbb{F}_{3^{2n}}$ and a cyclic cover of $\mathcal{X}$ of degree $m \in \{2,3,4,6\}$, we construct the corresponding $m$-twists over the…

In this note we present the main details of the construction of an elliptic curve over $\mathbb{Q}(u)$ with torsion $\mathbb{Z}/4\mathbb{Z}$ and rank 6. Previously only rank 5 examples for such curves were known.

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we…

Let $E$ be an elliptic curve over $\Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let…

In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding…

Consider the elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3 - 11x - 14$, which has complex multiplication by the order of conductor $2$ inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and…

We prove that there exist infinitely many elliptic curves over \Q with given modular invariant, and rank >=2. Furthermore, there exist infinitely many elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6 (resp.…

We find all elliptic curves defined over $\mathbb{Q}$ that have a rational point of order $N$, $N\ge 4$, and whose conductor is of the form $p^aq^b$, where p, q are two distinct primes, a, b are two positive integers. In particular, we…

We study the average rank of elliptic curves $E_{A,B} : y^2 = x^3 + Ax + B$ over $\mathbb{Q}$, ordered by the height function $h(E_{A,B}) := \text{max}(|A|, |B|)$. Understanding this average rank requires estimating the number of…

In this paper, we compute a formula for the $a$-number of certain hyperelliptic curves given by the equation $y^2= x^m+1$ for infinitely many values of $m$. The same question is studied for the curve corresponding to $y^2= x^m+x$.

Let $C_{m} : y^{2} = x^{3} - m^{2}x +p^{2}q^{2}$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p, q$ are distinct odd primes. We study the torsion part and the rank of $C_m(\mathbb{Q})$. More…

This article considers the family of elliptic curves given by $E_{pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod {40}$ and $ q \equiv 7 \pmod {40}$, then the…

By the Mordell-Weil theorem the group of Q(z)-rational points of an elliptic curve is finitely generated. It is not known whether the rank of this group can get arbitrary large as the curve varies. Mestre and Nagao have constructed examples…

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$,…

In this paper, we construct a family of elliptic curves of rank at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals $(p^3,q^3,r^3,q^3)$ not necessarily standing for the genuine sides of…