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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2022-08-09 12:48:02

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**History:**(expand/hide all)

- 2022-08-09 12:48:02 by Andrew Sutherland (Reviewed)
- 2020-03-31 13:53:06 by Andrew Sutherland (Reviewed)
- 2019-04-22 20:43:42 by Andrew Sutherland (Reviewed)
- 2019-03-21 23:42:59 by Andrew Sutherland
- 2019-03-21 22:48:34 by Andrew Sutherland
- 2018-07-26 19:28:35 by Andrew Sutherland (Reviewed)

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