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This chapter describes features provided for working with elliptic
curves in Magma. It contains basic functionality for curves over
fairly general fields, and also specialised functions for curves
over the rationals and number fields. There are separate chapters
dealing with specialised functions for curves over finite fields,
and over univariate function fields.
An elliptic curve E is the projective closure of
the curve given by the generalized Weierstrass equation
y2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6,
and is specified by the sequence [a1, a2, a3, a4, a6] of
coefficients, or alternatively by the two element sequence [a4, a6] 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 we implement 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 (over number fields),
and 3-descent and 4-descent (over the rationals). For curves over
the rationals, several aspects of the analytic theory (including modular
parametrizations and Heegner points) are implemented.
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, classifying the local structure of the special fibre at p
of the N'eron model of an elliptic curve E/Q.
This chapter, the first of three on elliptic curves, begins with a
treatment of the basics for curves over general fields: their construction,
their arithmetic and their basic properties. The chapter then presents
the wide range of techniques available for determining information about
the group of rational points for curves over Q and over number fields.
Specialised machinery provided for elliptic curves over finite fields
is described in Chapter ELLIPTIC CURVES OVER FINITE FIELDS, while elliptic curves over
function fields are discussed in Chapter ELLIPTIC CURVES OVER FUNCTION FIELDS.
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