3 edited answer to reflect edits in question

Finding all the integral points on an elliptic curve is a non-trivial computational problem. You say you are a "non-professional" so here is a non-professional answer: get hold of some mathematical software that does it for you (e.g. MAGMA), and then let it run until it either finds the answer or runs out of memory. Alternatively, do what perhaps you should have done at the start if you just have one curve and want to know the answer: post the equation of the curve, and hope that someone else does it for you. Here's another example of an algorithm currently used in these sorts of software (a Thue one was mentioned above but here's a different approach): find generators for the group (already computationally a bit expensive at times, depending on your luck and/or the size of sha), invoke Baker-like theorems saying "if the coordinates of the point are integral then it must be of the form sum_i n_i P_i with the n_i at most ten to the billionbillion", and then use clever congruence techniques to massively cut down the search space by giving strong congruences for all the n_i. Then just do a brute force search.

Whether or not this will work for you, I cannot say, because it all depends on how big the coordinates of your curve are. The only clue you give so far is that the conductor is "bigger than 130000" [Edit: that was written before the OP edited the question to tell us which curve he was interested in] which of course does not preclude it being bigger than 10^10^10. Also, you need an expert to decide which of the algorithms is best for you. I'd rather do a massive amount of arithmetic in a field of tiny discriminant than a small amount of arithmetic in a field whose discriminant is so large that I can't even factor it, for example.

So in short the answer is that you're probably not going to be able to do it with pencil and paper, but there are programs around that will do it, if all you want to know is the answer.

EDIT: you posted the equation of the curve. Magma V2.15-10 says the integral points are

[ <-23, -196>, <19, 182>, <61, 784>, <-191, 28>, <103, -1442>, <-19, -144>, <-67, 592>, <23, 242>, <-49, -454>, <-157, -742>, <817, 21196>, <521, 11364>, <3857, 200404>, <10687, -910154>, <276251, -118593646> ]

plus what you get if you change all the y's to -y's.

Finding all the integral points on an elliptic curve is a non-trivial computational problem. You say you are a "non-professional" so here is a non-professional answer: get hold of some mathematical software that does it for you (e.g. MAGMA), and then let it run until it either finds the answer or runs out of memory. Alternatively, do what perhaps you should have done at the start if you just have one curve and want to know the answer: post the equation of the curve, and hope that someone else does it for you. Here's another example of an algorithm currently used in these sorts of software (a Thue one was mentioned above but here's a different approach): find generators for the group (already computationally a bit expensive at times, depending on your luck and/or the size of sha), invoke Baker-like theorems saying "if the coordinates of the point are integral then it must be of the form sum_i n_i P_i with the n_i at most ten to the billion, and then use clever congruence techniques to massively cut down the search space by giving strong congruences for all the n_i. Then just do a brute force search.

Whether or not this will work for you, I cannot say, because it all depends on how big the coordinates of your curve are. The only clue you give so far is that the conductor is "bigger than 130000" which of course does not preclude it being bigger than 10^10^10. Also, you need an expert to decide which of the algorithms is best for you. I'd rather do a massive amount of arithmetic in a field of tiny discriminant than a small amount of arithmetic in a field whose discriminant is so large that I can't even factor it, for example.

So in short the answer is that you're probably not going to be able to do it with pencil and paper, but there are programs around that will do it, if all you want to know is the answer.

EDIT: you posted the equation of the curve. Magma V2.15-10 says the integral points are

[ <-23, -196>, <19, 182>, <61, 784>, <-191, 28>, <103, -1442>, <-19, -144>, <-67, 592>, <23, 242>, <-49, -454>, <-157, -742>, <817, 21196>, <521, 11364>, <3857, 200404>, <10687, -910154>, <276251, -118593646> ]

plus what you get if you change all the y's to -y's.

1

Finding all the integral points on an elliptic curve is a non-trivial computational problem. You say you are a "non-professional" so here is a non-professional answer: get hold of some mathematical software that does it for you (e.g. MAGMA), and then let it run until it either finds the answer or runs out of memory. Alternatively, do what perhaps you should have done at the start if you just have one curve and want to know the answer: post the equation of the curve, and hope that someone else does it for you. Here's another example of an algorithm currently used in these sorts of software (a Thue one was mentioned above but here's a different approach): find generators for the group (already computationally a bit expensive at times, depending on your luck and/or the size of sha), invoke Baker-like theorems saying "if the coordinates of the point are integral then it must be of the form sum_i n_i P_i with the n_i at most ten to the billion, and then use clever congruence techniques to massively cut down the search space by giving strong congruences for all the n_i. Then just do a brute force search.

Whether or not this will work for you, I cannot say, because it all depends on how big the coordinates of your curve are. The only clue you give so far is that the conductor is "bigger than 130000" which of course does not preclude it being bigger than 10^10^10. Also, you need an expert to decide which of the algorithms is best for you. I'd rather do a massive amount of arithmetic in a field of tiny discriminant than a small amount of arithmetic in a field whose discriminant is so large that I can't even factor it, for example.

So in short the answer is that you're probably not going to be able to do it with pencil and paper, but there are programs around that will do it, if all you want to know is the answer.