LMS Journal of Computation and Mathematics

Research Article

Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

Robert L. Millera1a2 p1

a1 Warwick Mathematics Institute Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom

a2 The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley CA 94720-5070, USA


We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.

(Received April 10 2011)

(Revised June 27 2011)

(Online publication November 2011)

2000 Mathematics Subject Classification

  • 11G40 (primary);
  • 14G10;
  • 11-04 (secondary)


p1 Current address: Quid, Inc., 733 Front Street, C1A, San Francisco CA 94111, USA (email: rmiller@quid.com)