This chapter describes features for working with elliptic curves in Magma. It contains basic functionality that is applicable to curves over fairly general fields. There are separate chapters describing features that are specific to
y2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6.
The curve is specified by the sequence of coefficients [a1, a2, a3, a4, a6]; the two element sequence [a4, a6] may be used instead when a1 = a2 = a3 = 0.
Elliptic curve functionality covers both elementary invariants of curves and arithmetic in the group of rational points, as well as higher level features for computing local invariants, heights, and Mordell--Weil groups for curves over Q, and for point counting and the determination of the group structure over Fq. The base ring of elliptic curves is currently restricted to fields. Curves over the rationals have special features to allow the construction of integral and minimal models, and for base change to finite fields, in acknowledgement of the integral structure over Z or Zp.
For curves over the rationals or over number fields there are routines for determining minimal models and an implementation of Tate's algorithm for determining Kodaira symbols and various local invariants. Algorithms for the computation of the Mordell--Weil group are heavily based on publications of John Cremona; see [Cre97] for details. There are also separate implementations of 2-descent for curves over number fields, and 3- and 4-descent for curves over the rationals. Additionally, several aspects of the analytic theory (including modular parametrisations and Heegner points) are implemented for curves over the rationals.
Elliptic curves are specialised forms of the more general curve and scheme types, and as such all functions which apply to these general types work on elliptic curves (although a few of them behave differently for elliptic curves). Some of these functions are described here, but not all of them --- refer to chapters SCHEMES (Schemes) and ALGEBRAIC CURVES (Curves) for descriptions of these functions, as well as an explanation of the relationships between points, point sets, and schemes. In particular, note that the parent of a point is a point set, and not the curve.
The name of the category of elliptic curves is CrvEll, with points of type PtEll lying in point sets of type SetPtEll. There is also the category SchGrpEll for subgroup schemes of elliptic curves, and a special category SymKod exists for the datatype of Kodaira symbols, which classify the local structure of the special fibre at p of the Néron model of an elliptic curve E/Q.
This chapter, the first of four on elliptic curves, contains a treatment of the basics for curves over general fields: their construction, their arithmetic, and their basic properties. Specialised machinery provided for elliptic curves over finite fields is described in Chapter ELLIPTIC CURVES OVER FINITE FIELDS; Chapter ELLIPTIC CURVES OVER Q AND NUMBER FIELDS presents the wide range of techniques available for determining information about the group of rational points for curves over Q and over number fields, while elliptic curves over function fields are discussed in Chapter ELLIPTIC CURVES OVER FUNCTION FIELDS.