On Elliptic Curves of Conductor N=PQ

We study elliptic curves with conductor N = pq for p and q prime. By studying the 2-torsion field we obtain that for N a product of primes satisfying some congruency conditions and class number conditions on related quadratic fields, any elliptic curve of conductor N has a rational point of order 2. By studying a minimal Weierstrass equation and its discriminant we obtain a solution to some Diophantine equation from any curve with conductor N = pq and a rational point of order 2. Under certain congruency conditions, this equation has no solutions, and so we conclude that in this situation there is no elliptic curve of conductor N with a rational point of order 2. Combining these two results, we prove that for a family of N = pq satisfying more specific congruency conditions and class number conditions on related quadratic fields, there are no elliptic curves of conductor N. We use a computer to find all N < 10^7 satisfying these conditions, of which there are 67. This work is similar to and largely inspired by past work on conductors p by Ogg [14, 15], Hadano [9], Neumann [13], Setzer [17], and Brumer and Kramer [4]