TITLE: Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with Sage
QUESTION [6 upvotes]: I'm trying to use Sage on-line,but I meet some trouble with the code of it.
I want to find the rational points on an ellipse curve,such as $$1 + 18 x + 81 x^2 + 44 x^3 = y^2,\tag1$$
I know that $(x,y)=(0,1)(1,\pm12)(-\frac{1}{11},0)$ are on the curve,but I don't know how to find more,so I try to solve it with the math-software Sage.
What should I input?Thanks in advance!
Edit: Multiply by $44^2$,we can difine the ellipse curve by this way,
E = EllipticCurve([0,81,0,44*18,44^2])
Then how to find the rational points on it?

REPLY [4 votes]: In case a curve has positive rank, you'd like to know its generators by E.gens().
In the case 
E = EllipticCurve([0,121,0,22*52,52^2]);

the rank is 1
sage: E.rank()
1

and its generator is 
sage: E.gens()
[(-104 : 260 : 1)]

In this case you can construct infinitely many points on E, by adding a torsion point and some multiple of the generator, for instance
sage: T = E(0,52)
sage: P = E(-104,260) 
sage: T + 2*P
(480/49 : 55796/343 : 1)
sage: T + 3*P
(-12662312/485809 : -65467251044/338608873 : 1)
sage: T - 5*P
(-70265016926482573/11385999920450401 : 5045808098009295785013222/1214945410097644049235601 : 1)

and so on.