The Pari-Script of the 3-Selmer Group of
an elliptic curve: y2 = x3 + b
Sungkon Chang
University of Georgia

I wrote a Pari-script which computes the 3-Selmer group of an elliptic curve given by the above equation. The script is based upon the algorithm introduced in Schaefer-Stoll's paper: How to do a p-descent on an elliptic curve link, Trans AMS. One may use this script together with Cremona's mwrank to compute the 3-part of the Tate-Shaferavich group.

Warning: It was written for the Gp-Pari version 2.1.5, and it may not run for a higher version as newer versions use different formats for the outputs of some commands. I am currently revising it for newer versions.

Report    Script

The program is run in the gp-calculator of the Pari system:
  • Run the gp-calculator in the directory where the script is placed.
  • In the gp-calculator, type \r selmer3333 to load the script.
  • Example: selmer(17,70).
    The above command computes the dimension of the 3-Selmer group of y2=x3+17,
    and 70 in the second argument is the level of p-adic precision.

For more explanation, refer to the report, and please, report bugs to (