Is there a web site where one can look up a Weierstrass equation, by discriminant say, or coefficients of some readily derivable "standard" form, and find the rank of its solutions over Q neatly listed, along with a set of generators and torsion group?
After several web searches, the nearest I've come is the Cremona Tables. But they list by conductor, whatever that is, and there seems no obvious way for an amateur such as myself to translate the data on that site into a form usable for the above mentioned purpose.
Failing that, I'd be content with an eay to use Mathematica or Sage package to achieve this, with idiot-proof instructions.
Right now, I'm especially interested in the equation $ x^2 + x (x + 1)^2 = y^2 $, which obviously has some rational solutions, and I'm pretty sure has positive rank over Q.
But the answer for that specific equation, although useful at present, obviously won't help with others that may interest me in future - "Teach a Chinaman to fish" and all that ..
P.S. If the Cremona Tables can easily be used to look up equations such as the example above, I'd very much appreciate a simple walkthrough, using the above as an example, and I think others would also find this useful.