In cryptographic applications, hyperelliptic curves of small genus have the advantage of providing a group of comparable size to that of elliptic curves, while working over a field of smaller size. Pairing-friendly hyperelliptic curves are those for which the order of the Jacobian is divisible by a large prime, whose embedding degree is small enough for pairing computations to be feasible, and whose minimal embedding field is large enough for the discrete logarithm problem in it to be difficult. We give a sequence of 𝔽 q -isogeny classes for a family of Jacobians of genus 2 curves over 𝔽 q , for q = 2 m , and the corresponding small embedding degrees. We give examples of the parameters for such curves with embedding degree k < (log q ) 2 , such as k = 8, 13, 16, 23, 26. For secure and efficient implementation of pairing-based cryptography on genus g curves over 𝔽 q , it is desirable that the ratio be approximately 1, where N is the order of the subgroup with embedding degree k . We show that for our family of curves, ρ is between 1 and 2. We also give a sequence of 𝔽 q -isogeny classes for a family of Jacobians of genus 2 curves over 𝔽 q for which the minimal embedding field is much smaller than the finite field indicated by the embedding degree k . That is, the extension degrees in this example differ by a factor of m , where q = 2 m , demonstrating that the embedding degree can be a far from accurate measure of security. As a result, we use an indicator to examine the cryptographic security of our family of curves.