bound for the rank of an elliptic curve


Let E/Q be an elliptic curveMathworldPlanetmath given by the equation:

E:y2=x(x-t)(x-s), with t,s

and suppose that E has s=m+a primes of bad reduction, with m and a being the number of primes with multiplicative and additive reduction respectively. Then the rank of E, denoted by RE, satisfies:


As an application of the theorem above, we can prove that E1:y2=x3-x has only finitely many rational solutions. Indeed, the discriminantPlanetmathPlanetmathPlanetmath of E1, Δ=64, is only divisible by p=2, which is a prime of (bad) multiplicative reduction. Therefore RE1=0. Moreover, the Nagell-Lutz theorem implies that the only torsion points on E1 are those of order 2. Hence, the only rational points on E1 are:



  • 1 James Milne, Elliptic Curves, online course notes.
Title bound for the rank of an elliptic curve
Canonical name BoundForTheRankOfAnEllipticCurve
Date of creation 2013-03-22 14:24:25
Last modified on 2013-03-22 14:24:25
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Theorem
Classification msc 14H52
Related topic ArithmeticOfEllipticCurves