Mathematical Research Letters

Volume 19 (2012)

Number 2

Isomorphism classes of elliptic curves over a finite field in some thin families

Pages: 335 – 343

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n2.a6

Authors

Javier Cilleruelo (Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid 28049, España.)

Igor E. Shparlinski (Department of Computing, Macquarie University, Sydney, NSW 2109, Australia.)

Ana Zumalacárregui (Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid 28049, España.)

Abstract

For a prime $p$ and a given square box, $\B$, we consider allelliptic curves $E_{r,s}:Y^2=X^3+rX+s$ defined over a field$\F_p$ of $p$ elements with coefficients $(r,s)\in\B$. Weobtain a nontrivial upper bound for the number of suchcurves which are isomorphic to a given one over $\F_p$, interms of the size of $\B$. We also give an optimal lowerbound on the number of distinct isomorphic classesrepresented.

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