Location of Repository

Computing all integer solutions of a genus 1 equation

By R.J. Stroeker and N. Tzanakis


The Elliptic Logarithm Method has been applied with great successto the problem of computing all integer solutions of equations ofdegree 3 and 4 defining elliptic curves. We extend this methodto include any equation f(u,v)=0 that defines a curve of genus 1.Here f is a polynomial with integer coefficients and irreducible overthe algebraic closure of the rationals, but is otherwise of arbitrary shape and degree.We give a detailed description of the general features of our approach,and conclude with two rather unusual examples corresponding to equationsof degree 5 and degree 9.Elliptic curve;Elliptic logarithm;Dophantine equation

OAI identifier:

Suggested articles



  1. (2000). A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function,
  2. (1992). A quantitative version of Runge’s theorem on diophantine equations,
  3. (1978). Algebraic Curves,
  4. (1933). Algebraic Functions,
  5. (1994). An algorithm for computing an integral basis in an algebraic number field,
  6. (1995). An algorithm for computing the Weierstrass normal form,
  7. (2000). Computing all integer solutions of a general elliptic equation, In: Algorithmic Number Theory
  8. (1994). Computing integral points on elliptic curves,
  9. (1994). Computing parametrizations of rational algebraic curves,
  10. (1970). Construction of rational functions on a curve,
  11. (2001). Height estimates for elliptic curves in short Weierstraß form over global fields and a comparison,
  12. (1992). Integer points on curves of genus 1,
  13. (1951). Introduction to the theory of algebraic functions of one variable,
  14. (2000). Irreducibility testing over local fields,
  15. (1991). Lectures on Elliptic Curves, London Math. Soc. Student Texts 24,
  16. (1995). Minorations de formes lin´ eaires de logarithmes elliptiques,
  17. (1998). On Mordell’s equation,
  18. (1997). On sums of consecutive squares,
  19. (1999). On the complexity of rational Puiseux expansions,
  20. (1976). On the difference of the Weil height and the N´ eron-Tate height,
  21. (1999). On the Elliptic Logarithm Method for Elliptic Diophantine Equations: Reflections and an improvement,
  22. (1995). On the sum of consecutive cubes being a perfect square,
  23. (1994). Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms,
  24. (1996). Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations.
  25. (1999). Solving elliptic diophantine equations: the general cubic case,
  26. (1990). The difference between the Weil height and the canonical height on elliptic curves,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.