Related papers: Heuristics on pairing-friendly elliptic curves

We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining…

We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves E_k: x^3 + y^3 = k of ranks 8, 9, 10, and 11 over Q. As a corollary we produce…

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with logarithmic Kirchhoff function. We establish the precise asymptotic formulas for the solution $u_\lambda(x)$ as $\lambda \to \infty$. Here, $\lambda > 0$ is the…

The problem of constructing elliptic curves suitable for pairing applications has received a lot of attention. To solve this, we propose a variant algorithm of a known method by Brezing and Weng. We produce new families of parameters using…

Given an elliptic curve $E$ and a finite Abelian group $G$, we consider the problem of counting the number of primes $p$ for which the group of points modulo $p$ is isomorphic to $G$. Under a certain conjecture concerning the distribution…

Miyaji, Nakabayashi, and Takano proposed the algorithm for the construction of prime order pairing-friendly elliptic curves with embedding degrees $k=3,4,6$. We present a method for generating generalized MNT curves. The order of such…

We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic…

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q…

Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a…

We study whether a complete family of pairing friendly elliptic curves has a \rho-value 1 or not. We show that, in some cases, \rho-values are not to be 1.

Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in…

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This…

We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only…

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree $\ell$ ($\ell$ different from the…

We show that the distribution of elliptic curves in isogeny classes of curves with a given value of the Frobenius trace $t$ becomes close to uniform even when $t$ is averaged over very short intervals inside the Hasse-Weil interval.

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where…

We provide an explicit classification of supersingular elliptic curves in characteristic~3 into isomorphism classes, and give explicit formulae for their point counts. This report was written specifically to support implementation of point…

We list the elliptic curves defined over $Q(\sqrt 5)$ with good reduction away from 2. There are 368 isomorphism classes. We give a global minimal model for each class.

An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $\phi:E\rightarrow E^\prime$ comes equipped with an invariant $\alpha_{\phi/K}$ that measures the valuation of the leading term of the formal group homomorphism…

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations.…