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How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.?

As a specific example, do there exist integers $x$ and $y$ such that $$ x^3+x^2y-y^3-y+3 = 0 ? $$

As noted in the previous Mathoverflow question Integer points of an elliptic curve (and in the comments to this question) there is a paper

Stroeker and de Weger's "Solving elliptic diophantine equations: The general cubic case."

(see also Stroeker, Roel, and Nikolaus Tzanakis. "Computing all integer solutions of a genus 1 equation." Mathematics of computation 72.244 (2003): 1917-1933. for higher degree equations)

that describes the algorithm, but the details are quite non-trivial to work out by hand, especially because I need to solve many equations in this form.

Interestingly, in the more difficult case of genus 2 equations there is a Magma code, see e.g.

https://math.stackexchange.com/questions/2700512/is-x5x5-y2-solvable-over-the-integers/2700727

and

Find all rational solutions of $x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1)$

that, for many specific equations (although not in general) able to automatically find all rational (and therefore all integer) solutions. Hence, it is a bit surprise for me that integer points on genus 1 curve (including Elliptic curve given by cubic equation in non-Weierstrass form) turned out to be more difficult for implementation.

(Remark: The question has been revised in the light of comments of Chris Wuthrich that you can find below).

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    $\begingroup$ You must be aware that your question is a duplicate of the question mathoverflow.net/questions/6676/… . $\endgroup$ Oct 29, 2021 at 12:58
  • $\begingroup$ Does this answer your question? Integer points of an elliptic curve $\endgroup$ Oct 29, 2021 at 13:07
  • $\begingroup$ I am aware of that question, and my question is not a duplicate but rather follow up. That question is about elliptic curves, and the first part of my question is how to check that a given genus 1 equation has rational point and is therefore an elliptic curve. The second part of my question is about non-Weierstrass form. $\endgroup$ Oct 29, 2021 at 13:40
  • $\begingroup$ Ok, I see, the second part is exactly what was asked in the other question (including non-Weierstrass) and the answer by Franz Lemmermayer and the link to Stroeker and de Weger's "Solving elliptic diophantine equations: The general cubic case." answers that question. I would rather want to see someone adding answers to the old question if there is more to say about it than answering here. $\endgroup$ Oct 29, 2021 at 17:21
  • $\begingroup$ To your first question. If there is a local obstruction for the existence of a rational point on your genus 1 curve, then you are done. If there are no local obstructions and you have searched for a while, then you will have problems deciding if there is a point and even more for finding one. You could calculate (by p-adic methods or Heegner points) the Tate-Shafarevich group of the Jacobian and that might tell you that there must be a point. $\endgroup$ Oct 29, 2021 at 17:29

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