show · rcs.source.g2c all knowls · up · search:

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'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 label "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 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.

• 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:math/9804069, MR:1717533] . There are 54 curves for which the Tamagawa number at 2 has not been computed.

Authors:
Knowl status:
• Review status: reviewed
• Last edited by Andrew Sutherland on 2019-05-03 18:36:54
Referred to by:
History: Differences