Genus 2 curves whose Jacobians have good reduction away from 2
I’ve been on a quest to find as many genus 2 curves $C/\mathbb{Q}$ as I can whose Jacobian has good reduction away from 2. This is still work in progress; so far we’ve found 512 such curves, including the 366 found by Smart. We are indebted to the LMFDB for inspiring this project.
Genus 2 curves
| File | Contents |
|---|---|
| genus2_good2.txt | A list of polynomials $f(x)$, each denoting a genus 2 curve given as a simplified model $y^2 = f(x)$. |
| genus2_good2_stats.txt | Each line is in the format $\texttt{D:N:r:T:f(x):A:S}$, where $D$ is the absolute discriminant, $N$ the conductor, $r$ the rank, $T$ the torsion subgroup, $f(x)$ a polynomial defining $C$, $A$ the automorphism group, and $S$ the Sato–Tate group. |
| genus2_good2_stats.pdf | A LaTeX-formatted summary of the various invariants (minimal discriminant, conductor, rank, and so on) for each curve $C/\mathbb{Q}$. For field systems we adopt the same notation as Smart. |
| genus2_good2_geometric_stats.pdf | A LaTeX-formatted summary of invariants of the 67 $\overline{\mathbb{Q}}$-isomorphism classes of the curves above (G2-invariants, geometric bad primes, geometric automorphism group, and so on). |
Abelian surfaces
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.
| File | Contents |
|---|---|
| abelian2_good2.pdf | A LaTeX-formatted summary of the various invariants (conductor, rank, endomorphism algebra, and so on) for each isogeny class of abelian surface $A/\mathbb{Q}$ found. |
| lfunctions2_good2.txt | Each line is in the format $\texttt{N:r:[L2,L3,…,L97]:S:m:c}$, where $N$ is the conductor, $r$ the rank, $L_2, L_3, \ldots, L_{97}$ the first few Euler factors, $m$ the number of known genus 2 curves $C/\mathbb{Q}$ whose Jacobian lies in this isogeny class, and $c$ the leading coefficient of the L-function. |
Notes
For details on how these curves and all their invariants were computed, see Chapters 5 and 6 of my thesis.
If you know of any examples of abelian surfaces good away from 2 not given in the tables above, please let me know. A £100 prize was offered for any new genus 2 curve whose Jacobian has good reduction away from 2, subject to terms and conditions.
Update (8 June 2026). The prize has been claimed by Raymond van Bommel, Céline Maistret, Jia Shi and Andrew V. Sutherland, who found ten new genus 2 curves — see their paper.
