Homework 7: Class Groups and Elliptic Curves

Due Wednesday, November 14

William Stein

Math 124 Harvard University Fall 2001

1. (10 points) For any negative discriminant D, let C_{D} denote the finite abelian group of equivalence classes of primitive positive definite quadratic forms of discriminant D. Use the PARI program forms.gp from lecture 24 (download it from my web page) to compute representatives for C_{D} and determine the structure of C_{D} as a produce of cyclic groups for each of the following five values of D:

D = −155, −231, −660, −12104, −10015.

2. (6 points) Draw a beautiful graph of the set E(ℝ) of real points on each of the following elliptic curves:

(i) y^{2} = x^{3} − 1296x + 11664,

(ii) y^{2} + y = x^{3} − x,

(iii) y^{2} + y = x^{3} − x^{2} − 10x − 20.

3. (4 points) A rational solution to the equation y^{2} − x^{3} = −2 is (3, 5). Find a rational solution with x ≠ 3 by drawing the tangent line to (3, 5) and computing the third point of intersection.