twocover-results

This repository stores the results of running an implementation of two-cover descent for genus 2 quintic curves on all 7692 such curves of rank 2 or 3 in the LMFDB's database. The code is here, and the theory and formulas are explained in the following paper: https://arxiv.org/abs/2009.10313

The data is stored in the json folder, which contains one JSON file for each curve, titled according to the curve's LMFDB label. The JSON files contain data in the following format:

• "label": the curve's LMFDB label
• "conductor": the curve's conductor
• "disc": the absolute value of the discriminant of the curve
• "coeffs": a list of the coefficients of an integral sextic polynomial f such that y^2 = f(x) defines the curve
• "pts": a list of the (projective) points on the curve, or null if the points have not been provably computed (conditional on GRH)
• "count": the number of points if computed
• "root": the chosen root of the above sextic
• "g": a list whose entries are lists of the coefficients of the factors of f(x)/(x - root) over Q
• "x-coords": a list of x-coordinates of points on the curve. May be null or incomplete if the points have not been provably computed.
• "verified": is the above list verified complete (conditional on GRH)?
• "obstruction_found": have we found an obstruction to the algorithm completing?
• "search_bound": how far have we searched before for rational points on the twists?
• "twists":
  • "coeffs": list of the coefficients of the twist parameter in the form "num/denom"
  • "found_pts": number of points that have been found by a search
  • "base_pt": chosen base point of the twist, or null if none found
  • "loc_solv": is the twist locally solvable?
  • "pts": list of coordinates of points once provably computed
  • "verified": have we verified that the list of points is complete (conditional on GRH)?
  • "g1":
    • "g": list of coefficients of the corresponding factor of f
    • "aInv": a-invariant data (the strings are in the form "num/denom")
    • "Ecov": equation data for map Ecov: E -> P^1
    • "rank": computed rank of MW group
    • "MW_proven": has the MW group been provably computed (conditional on GRH)?
    • "MW_orders": orders of generators of MW group
    • "gens": generators of MW group (str in the form "num/denom")
    • "gens_reduced": whether the MW generators have already been reduced
    • "x-coords": list of x-coordinates found by elliptic Chabauty
• "hasse_principle": do the curve's twists all appear to satisfy the Hasse principle?
• "runtime": estimated runtime in seconds taken so far to compute this data
• "exception": did an unhandled exception occur?

The logs folder stores log files that are produced when running the code and are useful for debugging.