  Pairing-based cryptographic schemes require so-called pairing-friendly
elliptic curves, which have special properties. The set of pairing-friendly
elliptic curves that are generated by given polynomials form a complete family.
Although a complete family with a $\rho$-value of 1 is the ideal case, there is
only one such example that is known, this was given by Barreto and Naehrig. We
prove that there are no ideal families with embedding degree 3, 4, or 6 and
that many complete families with embedding degree 8 or 12 are nonideal, even if
we chose noncyclotomic families.
