Conductor of elliptic curves with complex multiplication and elliptic curves of prime conductor

Elliptic Curves and Complex Multiplication

Elliptic Curves of Large Rank and Small Conductor

For r = 6, 7, . . . , 11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r = 6) to over 100 (for r = 10 and r = 11). We describe our search methods, and tabulate, for each r = 5, 6, . . . , 11, the five curves of lowest conductor, and (except for r = 11) also the five of lowest absolute disc...

Complex Multiplication Structure of Elliptic Curves

Let k be a finite field and let E be an elliptic curve over k. In this paper we describe, for each finite extension l of k, the structure of the group E(l) of points of E over l as a module over the ring R of endomorphisms of E that are defined over k. If the Frobenius endomorphism ? of E over k does not belong to the subring Z of R, then we find that E(l)$R R(?&1), where n is the degree of l o...

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

We find all elliptic curves defined over Q that have a rational point of order N, N ≥ 4, and whose conductor is of the form pq, 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.

Complex Multiplication Tests for Elliptic Curves

We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminant of the endomorphism ring of the curve.