As far as I know, there is no standard yet for cryptography based on the DLP over Jacobians of genus 2 curves. Yet, what can we say about the class number, and the discriminant of the complex multiplication field of curves that would be used in practice ? I don't think we have any competitive alternative to CM-methods to generate cryptographic curves, so the class number would be small, but what does small mean ? Like 5, 1000, or more ? How about the discriminant, what would be its typical size ? Would it more likely be smooth or have large prime divisors ?
What are the typical size of the discriminant and the class number of the CM-field of genus 2 curves that would be used in practice ?
