Source of Genus 2 Curve Data (reviewed)

The database of genus 2 curves was constructed by Andrew Booker, Jeroen Sijsling, Andrew Sutherland, John Voight, and Dan Yasaki. A detailed description of its construction can be found in [10.1112/S146115701600019X, arXiv:1602.03715, MR:3540958].

• Geometric invariants, minimal discriminants, automorphism groups, local solubility, number of rational Weierstrass points, 2-Selmer rank, torsion subgroup of the Jacobian, and squareness of Sha were computed using built-in Magma functions; code to reproduce these computations can be obtained on each curve's home page.

• The odd part of the conductor was computed using the Pari implementation of Qing Liu's algorithm [MR:1302311]. Euler factors at odd primes of bad reduction were computed using Magma.

• The 2-part of the conductor (originally computed analytically) has been rigorously verified by Tim Dokchitser and Christopher Doris [arXiv:1706.06162] using algebraic methods.

• The data on the endomorphism ring and geometric endomorphism ring has been rigorously certified by Davide Lombardo [arXiv:1610.09674] and by Edgar Costa, Nicolas Mascot, Jeroen Sijsling, and John Voight [arXiv:1705.09248], independently, by different methods. This rigorously confirms the Sato-Tate group computations.

• Tamagawa numbers were computed by Raymond von Bommel, as described in [arXiv:1711.10409], using the method of [arXiv:math/9804069, MR:1717533]. As of December, 2019, a complete set of Tamagawa numbers has been computed for all 66158 genus 2 curves in the database (prior to this data some Tamagawa numbers at 2 were missing).

• Rational points were computed using Magma's RationalPoints function for hyperelliptic curves (which incorporates code developed by Michael Stoll). In cases where the set of rational points has not been provably determined, this is indicated by the caption "Known rational points". In cases where the set of rational points has been provably determined (via some variant of Chabauty's method for genus 2 curves as implemented in Magma, also due to Michael Stoll), this is indicated by the caption "Rational points"; this applies to about half the curves in the database.

• Mordell-Weil generators were computed using Michael Stoll's new implementation of the MordellWeilGenus2 function in Magma, which applies a combination of several strategies to attempt to rigorously determine a basis for the Mordell-Weil group. In addition to the standard methods for computing the 2-Selmer rank of the Jacobian (and the 2-Selmer set of $\mathrm{Pic}1$), and brute force searching for points on the curve and its Jacobian, visualization methods are used to obtain better rank bounds (by determining rank bounds for quadratic twists), point searches are conducted on 2-covering spaces, and isogenies to other Jacobians, products of elliptic curves, and Weil restrictions of elliptic curves over quadratic fields are exploited. See the Magma handbook for further details. This functionality has made it possible to obtain a provably correct basis of the Mordell-Weil group in all but 89 cases, and in all but a handful of these cases (less than 10) the results are conditionally correct under the BSD conjecture (this is indicated whenever it applies).

• Regulators and real periods were computed by Raymond van Bommel (details of these computations will appear in a forthcoming manuscript). As of January, 2020 this data is available for all but two curves (but should be viewed as conditional on BSD in cases where the Mordell-Weil rank is not known).

• Approximate values of the leading coefficient of the L-functions of genus 2 curves were computed by Edgar Costa using software originally developed by David Platt and Andrew Booker. This was used in combination with other BSD invariants to compute an approximation to the analytic order of Sha in all but two cases where the regulator is not known.

• Cluster pictures for genus 2 curves were computed by Alex Best and Raymond van Bommel using the algorithms described in [arXiv:2007.01749].