Completeness of genus 2 curve data over Q\Q

The database contains all genus 2 curves over $\Q$ that can defined by an integral equation of the form $y^2+(h_3x^3+h_2x^2+h_1x+h_0)y=f_6x^6+f_5x^5+f_4x^4+f_3x^3+f_2x^2+f_1x+f_0,$ with $h_i\in\{0,1\}$ and $|f_i|\le 90$ and absolute discriminant $|\Delta|\le 10^6$. As explained in [MR:3540958, arXiv:1602.03715], it also includes all genus 2 curves of absolute discriminant $|\Delta|\le 10^6$ defined by an integral equation as above such that at least one of the following holds:

• $h_i\in \{0,1\}$ and $|f_i|\le 2 (3.51)^{6-i}$ for $0\le i\le 6$;
• $h_i\in\{0,1\}$ and $|f_i|\le (7.13)^{4-|i-3|}$ for $0\le i\le 6$;
• $h_i\in\{0,1\}$ and $\sum_{i=0}^6 \lceil \log_{10}(|f_i|+1)\rceil\le 10$.

To date all known genus 2 curves of absolute discriminant $|\Delta|\le 10^6$ have an integral model that satisfies these constraints, but it is heuristically likely that there exist curves that do not.