T w^2 = x^4 + 2x^2 + (1+4T^5)/5
This equation is easily written in Weierstrass canonical form if T is a square T^2, so I set about trying to find the rank, and some explicit generators, for its Jacobian in that case,
y^2 = x * ( x^2 - x + (1-T^10)/5 )
Over Christmas holiday I tried about a hundred values of T and ran the curves through existing elliptic curve software. In many cases I found points on the curves; but frequently there was either a nontrivial group Sha[2] or else the points were of large height, for searches on the homogeneous spaces found nothing. Also, it certainly did appear that there were points on each specialization T=t, but I could not find a point over Q(T) (that is, no parameterized solution to this equation). That would have been unusual.
By January I decided to seek outside help and received a number of helpful suggestions. The most dramatic, in retrospect, was a hint from Samir Siksek about how to carry out a second descent. This led to effective software to carry out this procedure. In time I was able to run this software for all the hundred curves in my sample set, and (1) greatly lowered the upper bounds for rank (2) found many more points on the curve.
By April I had enough data that it was possible to find a ,rational section on E, thus in particular giving at least one point on each curve. The same point was found nearly simultaneously, but in a rather different way, by Masato Kuwata.
Now I can present fairly complete data. For only 5 curves is the rank not completely known. Some 20 curves are missing one or more generators whose existence is only inferred from parity considerations. (Actually in no case have I proven that I have _generators_ for E, only for E/2E.)
There may yet be more patterns of interest in the data. If what you seek isn't in the tables, contact me -- I have other information stored which wasn't included in this summary table .
Some more comments about the data and its derivation are available in a letter I circulated about my computations.
Questions? Click here to send me mail at rusin@math.niu.edu