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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
- -
- curves over finite fields (Chapter ELLIPTIC CURVES OVER FINITE FIELDS),
- -
- curves over the rationals or number fields (Chapter ELLIPTIC CURVES OVER Q AND NUMBER FIELDS),
- -
- curves over univariate function fields (Chapter ELLIPTIC CURVES OVER FUNCTION FIELDS).
An elliptic curve E is the projective closure of
the curve given by the generalised Weierstrass equation
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.
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