Genus 2 curves whose Jacobian has good reduction away from 2
I've been on a quest to try to find as many genus 2 curves $C/\mathbb{Q}$ as I can, with the condition that the Jacobian of $C/\mathbb{Q}$ has good reduction away from 2! Whilst this is still work in progress, we've so far found 512 such curves (including 366 genus 2 curves good away from 2, found by SmartLink opens in a new window). We are indebted to the LMFDBLink opens in a new window for inspiring this project! Download links are given below:
| genus2_good2.txt | A text file giving a list of polynomials $f(x)$, which denotes a genus 2 curve given as a simplified model $y^2 = f(x)$. |
| genus2_good2_stats.txt | A text file where each line is in the format $\texttt{D:N:r:T:f(x):A:S}$. Here $D$ is the absolute discriminant, $N$ is the conductor, $r$ is the rank, $T$ is the torsion subgroup, $f(x)$ a polynomial defining $C$, $A$ is the automorphism group, and $S$ is the Sato-Tate group. |
| genus2_good2_stats.pdf | A pdf file giving a LaTeX formatted summary of the various invariants (minimal discriminant, conductor, rank, etc.) for each curve $C/\mathbb{Q}$. For field systems, we adopt the same notation as given in SmartLink opens in a new window. |
| genus2_good2_geometric_stats.pdf | A pdf file giving a LaTeX formatted summary of various invariants of the 67 $\overline{\mathbb{Q}}$-isomorphism classes of the genus 2 curves found above (G2-invariants, geometric bad primes, geometric automorphism group, etc.). |
Furthermore, by including products of elliptic curves $E/\mathbb{Q}$ good outside 2, and Weil restrictions of elliptic curves over quadratic fields $K$ good outside 2, we've found a total of 234 isogeny classes of abelian surfaces with good reduction away from 2.
| A pdf file giving a LaTeX formatted summary of the various invariants (conductor, rank, endomorphism algebra, etc.) for each isogeny class of abelian surface $A/\mathbb{Q}$ found. | |
| lfunctions2_good2.txt | A text file where each line is in the format $\texttt{N:r:[L2,L3,...,L97]:S:m:c}$. Here $N$ is the conductor, $r$ is the rank, $L_2, L_3, \dots, L_{97}$ the first few Euler factors, $m$ the number of known genus 2 curves $C/\mathbb{Q}$ whose Jacobian lies in this isogeny class, and the leading coefficient $c$ of the L-function. |
If you know of any examples of abelian surfaces good away from 2 not given in the above tables, then please let me know! In particular, if you can find any new genus 2 curve whose Jacobian has good reduction away from 2, you'll win £100 (subject to terms and conditions)!
