The auxiliary file attached to this article is a package for PARI written by S. J. Miller to compute the analytic rank of a curve. The instructions on how to use it at the command line in PARI follow:
to load in the program arank.gp for command line usage:
årank.gp
now in PARI let E be an elliptic curve:
E = ellinit([0,0,1,-7,6])
to find the rank simple type at the command prompt
ellarank(E)
here are some curves with given ranks:
rank 3: [0,0,1,-7,6]
rank 4: [1,-1,0,-79,289]
rank 5: [0,0,1,-79,342]
rank 6: [0,0,1,-7077,235516]
If the conductor is large, the program takes a bit longer to run. For example, consider the following
E5 = ellinit([0,0,0,-15823,767122])
ellarank(E5)
It will see the rank is 5, but it will take a noticeable amount of time. 5 points on curve are (81,130) (83,160) (74,38) (71,40) (69,62) The height matrix of these points is approximately 32.5.
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