On the equation

  title={On the equation},
  author={Andrew Bremner and John W. Cassels},
  journal={Mathematics of Computation},
Generators are found for the group of rational points on the title curve for all primes p 5 (mod 8) less than 1,000. The rank is always I in accordance with conjectures of Selmer and Mordell. Some of the generators are rather large. 

On a class of elliptic curves with rank at most two

In this note we consider the elliptic curves y 2 = x 3 + px defined over Q for primes p satisfying p ? 1 (mod 8), and review some of their properties. We then compute and list (in the supplement)

The Group of Rational Points

In this chapter we will prove Mordell’s theorem that the group of rational points on a non-singular cubic is finitely generated. There is a tool used in the proof called the height. In brief, the

Second p-descents on elliptic curves

An algorithm is described which computes the set of D in the Shafarevich-Tate group such that pD = C and obtains explicit models for these D as curves in projective space, which leads to a practical algorithm for performing explicit 9-descents on elliptic curves over Q.

Arithmetic on curves

We are constantly discovering new ways of understanding algebraic curves and their arithmetic properties. Questions about ‘rational points’—the interplay of arithmetic and algebra—have fascinated

Three points of great height on elliptic curves

We give three elliptic curves whose generators have great height, demonstrating along the way a moderately efficient method for finding such points.

Elliptic Curves over Local Fields

In this chapter we study the group of rational points on an elliptic curve defined over a field which is complete with respect to a discrete valuation. We start with some basic facts concerning

On Diophantine equations and nontrivial Racah coefficients

Some families of zeros of weight‐1 6j coefficients are given, each in terms of four parameters. They arise from a geometrical investigation of certain Diophantine equations. Some general remarks on

The Formal Group of an Elliptic Curve

Let E be an elliptic curve. In this chapter we study an “infinitesimal” neighborhood of E centered at the origin O.

Ranks of elliptic curves y^2=x^3\pm4px

Suppose that y 2 = x 3 +4px, y 2 = x 3 4px are elliptic curves where p is an odd prime. Then we treat ranks of these two curves according to the condition of prime number p. Mathematics Subject



Notes on elliptic curves. II.

----------------------------------------------------Nutzungsbedingungen DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung