Definition
Let be a field and let E denote an elliptic curve in Weierstrass form over . Then the following curve can be obtained:
where the curve has discriminant
and (0,0) has order 3. Before proving this, note that if the characteristic of, say q, is 2 modulo 3, then the curve has 3 points of order 3; and if q is 1 modulo 3, there are 8 points of order 3.
To prove that has order 3, it is sufficient to show that using the elliptic curve group law.
(i) Compute : if is the given curve and the point, then . So, in this case, since, then .
(ii) Compute : it can be done using the tangent and chord method, that is, first construct the line through the general point and find the other intersection point with the curve.
Let be the tangent to the curve at . Now, to find the points of intersection between the curve and the line, replace every by in the curve:
iff
The roots of this equation are the x-coordinates of and, so:
Then comparing the coefficients:
and ( and ).
Note that are known and by the implicit function theorem (which is the slope of the line ). Then, and P would be known. This is a general method to compute .
So applying it to P=(0,0):
since (note cannot be zero, otherwise the denominator would vanish at and the curve would be singular),
then where
Thus, , that is (0,0) has order 3 over .
Now, to obtain the Hessian curve, it is necessary to do the following transformation:
First let denote a root of the polynomial T3 − δT2 + δ2T/3 + a32 = 0.
where is determined from the formula:
=1/3((-27a32)1/3+)
Note that if has characteristic q ≡ 2 (mod 3), then every element of has a unique cube root; otherwise, it is necessary to consider an extension field of K.
Now by defining the following value another curve, C, is obtained, that is birationally equivalent to E:
:
which is called cubic Hessian form (in projective coordinates)
:
in the affine plane ( satisfying and ).
Furthermore, D3≠1 (otherwise, the curve would be singular)
Starting from the Hessian curve, a birationally equivalent Weierstrass equation is given by
under the transformations:
and
where:
- =/
- =/
Read more about this topic: Hessian Form Of An Elliptic Curve
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)