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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 [MR:3540958, arXiv:1602.03715] .

  • 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 in Magma; code to reproduce these computations can be obtained on each curve's home page.

  • Rational points were computed using Magma code provided by Michael Stoll. In cases where the set of rational points has not been provably determined, this is indicated by the label "known rational points". In cases where the set of rational points has been provably determined (via some variant of Chabauty's method implemented in Magma), this is indicated by the label "all rational points"; this applies to about half the curves in the database.

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

  • All L-function computations are conditional on the assumption that the L-function lies in the Selberg class (in particular, that it has a meromorphic continuation to $\C$ and satisfies a functional equation). This also applies to the Euler factor at 2 for curves with bad reduction at 2.

  • Subject to the assumption that the L-function lies in the Selberg class, the root number has been rigorously computed and the analytic ranks are rigorous upper bounds. For 99 percent of the curves in the database the Mordell-Weil rank of the Jacobian has been rigorously computed using Magma code provided by Michael Stoll; in every case this matches the listed analytic rank.

  • The data on the 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.

  • The Tamagawa numbers were computed by Raymond von Bommel, as described in [arXiv:1711.10409] , using the method of [arXiv:9804069, MR:1717533] . There are 54 curves for which the Tamagawa number at 2 has not been computed.

  • Isogeny class identifications are based on comparing L-functions and comparing Euler factors at good primes up to $2^{20}$. Jacobians that are not identified as being isogenous are provably non-isogenous. In cases where Jacobians are identified as belonging to the same isogeny class, this identification is heuristic, but in principle it can be made rigorous in any particular case via a Faltings-Serre argument.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew V. Sutherland on 2018-07-26 18:44:09.424000