On elliptic curves whose conductor is a product of two prime powers

TL;DR

This paper classifies elliptic curves over rationals with a rational point of order N (N≥4) and conductors composed of two prime powers, proving Szpiro's conjecture for these cases.

Contribution

It explicitly finds all such elliptic curves with conductors as products of two prime powers and confirms Szpiro's conjecture for them.

Findings

All elliptic curves with specified properties are classified.

Szpiro's conjecture holds for these elliptic curves.

The structure of these elliptic curves is characterized in detail.

Abstract

We find all elliptic curves defined over $\mathbb{Q}$ that have a rational point of order $N$, $N\ge 4$, and whose conductor is of the form $p^aq^b$, where p, q are two distinct primes, a, b are two positive integers. In particular, we prove that Szpiro's conjecture holds for these elliptic curves.