Looking up the Mordell-Weil rank and generator(s) of a Weierstrass Equation

This is a very old question, and although the old answers are okay, there's now a great source to get information about specific elliptic curves, with multiples ways to search. This includes Cremona's tables, vastly extended. It's the L-Series and Modular Forms Database (LMFDB).

An alternative to Sage is Pari-GP. You initialize the curve with the vector $[a_1,a_2,a_3,a_4,a_6]$ such that $$ y^2 + a_1xy + a_3y = x^2 + a_2x^2+a_4x +a_6 $$ it identifies it in Cremona's Database giving directly it's name in the database, the minimal model, the generators of the free part in the minimal model and the change of variable to obtain it:

   E = ellinit( [0,0,3,1,0] ); ellidentify(E) 
     [["40a3", [0, 0, 0, -2, 1], []], [1, -1, 0, 0]]

meaning that the Cremona name is 40a3, it's minimal model is $[0,0,0,-2,1]$ that is: $$ Y^2 = X^3 -2X +1 $$ the $mathtt{[]}$ meaning the rank is zero and finally that you obtain the minimal model making the change of variable $$(x,y) = (u^2 X + r, u^3Y+s u^2 X+r)$$ in this case: $(u,v,r,s) = [1,-1,-0,0]$ that is: $x = X-1, y = Y$.

Here's how to this in Sage:

sage: var('x,y')
(x, y)
sage: E = EllipticCurve(y^2 == x^2 + x*(x+1)^2); E
Elliptic Curve defined by y^2 = x^3 + 3*x^2 + x over Rational Field
sage: E.rank()
0
sage: E.conductor()
40
sage: E.gens()
[]
sage: E.torsion_points()
[(-1 : -1 : 1), (-1 : 1 : 1), (0 : 0 : 1), (0 : 1 : 0)]

or see the published worksheet.

If you had wanted to use Cremona's tables, you would have use some software (or by hand) to find the unique minimal Weierstrass model for your curve with $a_1,a_3 in {0,1}$ and $a_2in {0,-1,1}$, then look up that equation in his tables (some of the tables have the equations for curves in this form in the first column).

sage: E.minimal_model().a_invariants()
(0, 0, 0, -2, 1)

You can also look at my database by conductor (way more curves) or possibly this testing database. Lookup by equation isn't supported in my database yet.

By the way, in Sage, type EllipticCurve? for lots of help or see the documentation and also the documentation on databases.

Sage has excellent documentation (short and long), and can be used online. I first used Sage when trying to find the rational points on a specific elliptic curve. It took me less than an hour to:

1. find the website
2. find and read the relevant instructions on the website
3. write my first Sage notebook on the website
4. find out the answer to my original question

Needless to say I learned much more during that short time than finding the answer. So why don't you go ahead and try Sage here. BTW Sage code for your problem was supplied on MO several times (look for user William Stein!). Also, you might find this link useful.