TITLE: How to find all integer points on an elliptic curve?
QUESTION [10 upvotes]: How can I determine the integer points of a given elliptic curve if I know its rank and its torsion group? 
I read same basic books on elliptic curves but as a non-professional I didn't understand everything. Is it true that if rank is 0 and torsion group is isomorphic to a group of order $n$ then the number of integer points is $n-1$? And what is a good reference to learn to determine the integer points if the rank is positive? 
I tried to read the book Rational Points on Elliptic Curves but I didn't found an explicit algorithm. I just heard something like take some point and use group law to find the rest. But how can I be sure that I have found every point?
The curve I had on my mind is $2x^3 + 385x^2 + 256x - 58195 = 3y^2$. I'm not even sure if this is an elliptic curve. I mean why it is projective and why it is isomorphic to a closed subvariety of $\mathbb{P}_{\mathbb{Q}}^2$? And why it contain the priviledged rational point $(0,1,0)$?

REPLY [14 votes]: The following Sage code (which I ran with Sage 4.2.1) produces the solutions (and agrees with Magma!):
E = EllipticCurve([0, 1155,0,4608,-6285060])  
E1 = E.minimal_model()
pts1 = E1.S_integral_points([2,3])
iso = E1.isomorphism_to(E)
pts = [iso(P) for P in pts1]
solutions = [(x/6,y/18) for (x,y,z) in pts]
solutions = [(x,y) for (x,y) in solutions if x.is_integral() and y.is_integral()]
solutions

Before the first line I used pencil-and-paper to scale the equation to be in Weierstrass form (monic in both X and Y) which involves new variables 6X and 18Y;  the solutions a rescaled at the end.  In fact we found all S-integral solutions with S={2,3}, i.e. all solutions which are integral except at 2 and 3.  (There are 32 of these, of which only 15 are really integral).
John Cremona