Exercise Set 8:

Elliptic Curves, part 1

Math 414, Winter 2010, University of Washington

Due Wednesday, March 3, 2010

1. Write down an equation y^{2} = x^{3} + ax + b over a field K such that −16(4a^{3} +27b^{2}) = 0. Precisely what goes wrong when trying to endow the set E(K) = {(x, y) ∈ K × K : y^{2} = x^{3} + ax + b} ∪ {O} with a group structure?

2. One rational solution to the equation y^{2} = x^{3} − 11 is (3, 4). Find a rational solution with x ≠ 3 by drawing the tangent line to (3, 4) and computing the second point of intersection.

3. Let E be the elliptic curve over the finite field K = ℤ/7ℤ defined by

the equation

y^{2} = x^{3} + x.

(a) List all 8 elements of E(K).

(b) Is the finite abelian group E(K) cyclic or not?