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Function Field






The torsion or division polynomials are the polynomials
or functions defining the subschemes of n-torsion points on the
elliptic curve.  The affine equation order of the function field is
the natural home of the torsion polynomials.
In order to obtain the division polynomial as a univariate polynomial,
the syntax DivisionPolynomial is used; to obtain the same as a
function on the curve, the syntax DivisionFunction is used.




Subsections



Function Field and Polynomial Ring





FunctionField(E) : CrvEll -> FldFun



The function field of the elliptic curve.



EquationOrder(E) : CrvEll -> RngOrdFun



The affine equation order of the elliptic curve.



PolynomialRing(E) : CrvEll -> RngMPol, Map



Returns a bivariate polynomial ring over the coefficient ring of
E.  This ring is stored with E and is the parent of the various
multivariate polynomials associated with E, such as the division
polynomials and the "omega" polynomial of any isogeny with domain
E.  The second return value is the map to the EquationOrder.



DefiningPolynomial(E) : CrvEll -> RngMPolElt


DefiningEquation(E) : CrvEll -> RngMPolElt



The affine defining polynomial for E, as an element of
PolynomialRing(E).



Torsion Polynomials





DivisionFunction(E, n) : Fld, RngIntElt -> RngFunOrdElt



The n-th division polynomial as an element of the affine equation
order of the function field of E.



DivisionPolynomial(E, n) : CrvEll, RngIntElt   -> RngUPolElt, RngUPolElt, RngUPolElt


DivisionPolynomial(E, n, g) : CrvEll, RngIntElt, RngUPolElt   -> RngUPolElt, RngUPolElt, RngUPolElt



Given an elliptic curve E and an integer n, returns the
n-th division polynomial as a univariate polynomial over the
base ring of E.  If n is even, this polynomial has multiplicity
two at the nonzero 2-torsion points of the curve.
The second return value is the n-th division polynomial, divided
by the univariate 2-torsion polynomial if n is even.
The third argument is the cofactor, equal to the univariate
2-torsion polynomial if n is even, and 1 otherwise.


If a polynomial is passed as a third argument, the division
polynomial is computed efficiently modulo that polynomial.



TwoTorsionPolynomial(E) : CrvEll -> RngMPolElt



Returns the multivariate 2-torsion polynomial 2y + a_1x + a_3
as an element of PolynomialRing(E), where the equation
of the curve is

y^2 + a_1*xy + a_3*y = x^3 + a_2*x^2 + a_4*x + a_6.







Example CrvEll_DivisionPolynomial (H85E14)

Let E be an elliptic curve over a finite field K.
The following code fragment illustrates the relationship between
the roots in K of the n-th division polynomial for E,
and the x-coordinates of the points of n-torsion on E.







> K := GF(101);
> E := EllipticCurve([ K | 1, 1]);
> Roots(DivisionPolynomial(E, 5));
[ <86, 1>, <46, 1> ]
> [ P : P in RationalPoints(E) | 5*P eq E!0 ];
[ (86 : 34 : 1), (0 : 1 : 0), (46 : 25 : 1), (86 : 67 : 1), (46 : 76 : 1) ]


A point worth noting is that even if the roots of the division polynomial
lie in the field, the corresponding points may not, instead lying in a
quadratic extension.





> Roots(DivisionPolynomial(E, 9));
[ <4, 1>, <17, 1>, <28, 1>, <34, 1>, <77, 1> ]
> Points(E, 4);
[]
> K2<w> := ext<K | 2>;
> Points(E(K2), 4);
[ (4 : 82*w + 57 : 1), (4 : 19*w + 44 : 1) ]
> Order($1[1]);      
9





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