In 2010, Freeman, Scott, and Teske published a well-known taxonomy compiling the best known families of pairing-friendly elliptic curves. Since then, the research effort mostly shifted from the generation of pairing-friendly curves to the improvement of algorithms or the assessment of security parameters to resist the latest attacks on the discrete logarithm problem. Consequently, very few new families were discovered. However, the need of pairing-friendly curves of prime order in some new applications such as SNARKs has reignited the interest in the generation of pairing-friendly curves, with hope of finding families similar to the one discovered by Barreto and Naehrig. Building on the work of Kachisa, Schaefer, and Scott, we show that some particular elements of quadratic extensions of a cyclotomic field generate families of pairing-friendly curves with small parameters. By exhaustive search among these elements, we discovered new families of curves of embedding degree $k=20$, $k=22$ and $k=28$. We provide an open-source SageMath implementation of our technique. We obtain curves of cryptographic size from our new families and we give a proof-of-concept SageMath implementation of a pairing on some new curves.