The number field sieve, an algorithm with major applications in cryptography, uses the elliptic curve method of factorization (ECM) as a building block. It is a motivation to search parameterizations of infinite families of elliptic curves defined over a given number field K with exceptional image of-adic Galois representation. This boils down to making the complete list of finite indexed subgroups H of GL(2,Z_l) of genus 0 and 1 and computing K-rational points on the modular curve X_H. Fast algorithms for this latter step are proposed in [SZ17] and [RZB15]. We measure the consequences on ECM of the newly found families using a heuristic of Montgomery.