Introduction

This chapter deals with functionality for elliptic curves, which is available over arbitrary fields, with specific features available for curves over the rationals and number fields, and finite fields.

An elliptic curve $E$ is the projective closure of the curve given by the Weierstrass equation

\[y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6\]

specified by the list of coefficients [a1, a2, a3, a4, a6]. If $a_1 = a_2 = a_3 = 0$, this simplifies to

\[y^2 = x^3 + a_4 x + a_6\]

which we refer to as a short Weierstrass equation and which is specified by the two element list [a4, a6].