Homework 7: Class Groups and Elliptic Curves
DUE WEDNESDAY, NOVEMBER 14

William Stein

Date: Math 124 $\quad$ HARVARD UNIVERSITY $\quad$ 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 $E(\mathbb{R})$ 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 $x\neq 3$ by drawing the tangent line to $(3,5)$ and computing the third point of intersection.