Solutions to Elliptical Curve Problems

Construct the line L through any two points P1 and P2 such that they intersect a third point P3, by direct calculation or using Bezout's theorem:

Soln.


1) Sketch more of the elliptic curve (2) such that the section is shown for x = 4, y = ?


The y-coordinate occurs at: y (4)    =  [(4)³     –       (4)  +  1 ] ^½     =  [61 ] ^½   =  7.8

2)  Use the short Weierstrass form to generate another elliptic curve and graph it. Then obtain the discriminant and ensure it is non-vanishing. Thence obtain h(E).

The short Weierstrass form is:  y²  =   x³     + Ax + B


Let A = -2   and    B = 10   then we will generate:

y²  =   x^{3    -}- 2x + 10

The equation when graphed appears:

Then the discriminant :  

D  = -16 (4 A³   +  27 B² ) =   -16[( 4 (-2)³    + 27(10)²] = 

[ 512  +  (-16)2700 ] =    [512 -  43200]  = -42688

h (E) =   max (4 |A|³ ,   27 B²) =   (4 |-2|³ ,   27 (10)²) =  (32,  2700)